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Algorithm for the Proof of Theorem 11
Algorithm for equality languages
Algorithm for reducts of unary structures
  • Peter JonssonPeter Jonsson
  • Victor LagerkvistVictor Lagerkvist
We study the fine-grained complexity of NP-complete, infinite-domain constraint satisfaction problems (CSPs) parameterised by a set of first-order definable relations (with equality). Such CSPs are of central importance since they form a subclass of any infinite-domain CSP parameterised by a set of first-order definable relations over a relational structure (possibly containing more than just equality). We prove that under the randomised exponential-time hypothesis it is not possible to find c>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > 1$$\end{document} such that a CSP over an arbitrary finite equality language is solvable in O(cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(c^n)$$\end{document} time (n is the number of variables). Stronger lower bounds are possible for infinite equality languages where we rule out the existence of 2o(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{o(n \log n)}$$\end{document} time algorithms; a lower bound which also extends to satisfiability modulo theories solving for an arbitrary background theory. Despite these lower bounds we prove that for each c>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > 1$$\end{document} there exists an NP-hard equality CSP solvable in O(cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(c^n)$$\end{document} time. Lower bounds like these immediately ask for closely matching upper bounds, and we prove that a CSP over a finite equality language is always solvable in O(cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(c^n)$$\end{document} time for a fixed c, and manage to extend this algorithm to the much broader class of CSPs where constraints are formed by first-order formulas over a unary structure.
  • Amihood AmirAmihood Amir
  • Ayelet ButmanAyelet Butman
  • Gad M. LandauGad M. Landau
  • [...]
  • Dina SokolDina Sokol
A tandem repeat is an occurrence of two adjacent identical substrings. In this paper, we introduce the notion of a double string, which consists of two parallel strings, and we study the problem of locating all tandem repeats in a double string. The problem introduced here has applications beyond actual double strings, as we illustrate by solving two different problems with the algorithm of the double string tandem repeats problem. The first problem is that of finding all corner-sharing tandems in a 2-dimensional text, defined by Apostolico and Brimkov. The second problem is that of finding all scaled tandem repeats in a 1d text, where a scaled tandem repeat is defined as a string UU′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$UU'$$\end{document} such that U′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U'$$\end{document} is discrete scale of U. In addition to the algorithms for exact tandem repeats, we also present algorithms that solve the problem in the inexact sense, allowing up to k mismatches. We believe that this framework will open a new perspective for other problems in the future.
We give better approximation ratios for two Steiner Tree variants by combining known algorithms: the optimum 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{3}$$\end{document}-decomposition and iterative randomized rounding. The first problem is Steiner Tree with minimum number of Steiner points and bounded edge length problem (SMT-MSP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{SMT-MSP}}$$\end{document}). The input consists of a set of terminals R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{R}}$$\end{document} in the Euclidean space R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{\mathbf{2}}$$\end{document}. A feasible solution is a Steiner tree τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\tau }}$$\end{document} spanning R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{R}}$$\end{document} with Steiner points S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{S}}$$\end{document} such that every edge in τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\tau }}$$\end{document} has length at most 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{1}$$\end{document}. The objective is to minimize S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt{S}$$\end{document}. Previously, the best approximation ratio for SMT-MSP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{SMT-MSP}}$$\end{document} was 1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{1} +$$\end{document}ln(4)+ϵ≈2.386\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbf{4}) + {\varvec{\epsilon }} \approx \mathbf{2.386}$$\end{document}. We present a polynomial time algorithm with ratio 2.277\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{2.277}$$\end{document}. The second problem is Steiner Tree in quasi-bipartite graphs. It is a Steiner Tree problem on graph G=(V,E,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{G}}=({\varvec{V}},{\varvec{E}},{\varvec{c}})$$\end{document} with terminal set R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{R}}$$\end{document} when the edge set E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{E}}$$\end{document} does not include any edge between two vertices in V\R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{V}} \setminus {\varvec{R}}$$\end{document}. The best-known approximation ratio for this problem is 7360\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\mathbf{73}}{\mathbf{60}}$$\end{document}, We improve this ratio to 298245\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mathbf{298}}{\mathbf{245}}$$\end{document}.
An unbalanced 2P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2P_3$$\end{document}
The graph corresponding to the word w=aababbab\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w = aababbab$$\end{document}
An infinite antichain of quasi-chain graphs
The double-chain graph D3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_3$$\end{document}
The class of quasi-chain graphs is an extension of the well-studied class of chain graphs. This latter class enjoys many nice and important properties, such as bounded clique-width, implicit representation, well-quasi-ordering by induced subgraphs, etc. The class of quasi-chain graphs is substantially more complex. In particular, this class is not well-quasi-ordered by induced subgraphs, and the clique-width is not bounded in it. In the present paper, we show that the universe of quasi-chain graphs is at least as complex as the universe of permutations by establishing a bijection between the class of all permutations and a subclass of quasi-chain graphs. This implies, in particular, that the induced subgraph isomorphism problem is NP-complete for quasi-chain graphs. On the other hand, we propose a decomposition theorem for quasi-chain graphs that implies an implicit representation for graphs in this class and efficient solutions for some algorithmic problems that are generally intractable.
Given an n-vertex m-edge graph G of clique-width at most k, and a corresponding k-expression, we present algorithms for computing some well-known centrality indices (eccentricity and closeness) that run in O(2O(k)(n+m)1+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(2^{{\mathcal {O}}(k)}(n+m)^{1+\epsilon })$$\end{document} time for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document}. Doing so, we can solve various distance problems within the same amount of time, including: the diameter, the center, the Wiener index and the median set. Our run-times match conditional lower bounds of Coudert et al. (SODA’18) under the Strong Exponential-Time Hypothesis. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using O(klog2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(k\log ^2{n})$$\end{document} bits per vertex and constructible in O~(k(n+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {O}}(k(n+m))$$\end{document} time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an O~(kn2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {O}}(kn^2)$$\end{document}-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k, being given a k-expression. This partially answers an open question of Kratsch and Nelles (STACS’20). Our algorithms work for graphs with non-negative vertex-weights, under two different types of distances studied in the literature. For that, we introduce a new type of orthogonal range query as a side contribution of this work, that might be of independent interest.
An illustration of nice 2-clique path decomposition
(a) A planar graph G of maximum degree 4, (b) the intermediate rectilinear embedding G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document} of G, (c) the unit disk graph GC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{{\mathcal {C}}}$$\end{document}; the black disks are corresponding to the grid vertices of G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document}, the red disks are Steiner disks and the blue disks are the terminal disks (Color figure online)
An illustration of grid labelling. The blue disks are terminals, and the red and black disks are chosen Steiner vertices and not-chosen non-terminal vertices, respectively (Color figure online)
An illustration of the component contraction; (a) red disks are Steiners and blue disks are terminals; (b) red vertices are Steiner vertices and blue vertices are contracted terminal components (Color figure online)
(a) The schematic diagram of the cells, after adjusting the distance between adjacent cells which is 2+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+\epsilon $$\end{document}. The red disks inside each cells, are the coordinates where the center of the Steiner disk of radius 1 will be placed. The diagonal chains consisting of terminal disks of radius κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}, are connecting the cells diagonally. (b) The small black dots inside each cell are extra terminals of radius δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}. Consider a cell S[x, y]. The shaded grey disks are the potential disks and the shaded red disk is chosen in the solution from D[x, y] (Color figure online)
We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\subseteq V(G)$$\end{document} of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V\R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\setminus R$$\end{document}. The vertices of R are referred to as terminals and the vertices of V(G)\R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G)\setminus R$$\end{document} as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in nO(t+k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{O(\sqrt{t+k})}$$\end{document} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2O(k)nO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{O(k)}n^{O(1)}$$\end{document}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs Fomin (Discret. Comput. Geometry 62(4):879–911, 2019). We mention that the algorithmic results can be made to work for Steiner Tree problem on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree problem on disk graphs parameterized by k, is W[1]-hard.
A 2-dimensional lattice animal is a set of edge-connected cells on some lattice. In this paper, we address the problem of counting minimum-perimeter lattice animals, that is, animals that have the minimum possible perimeter of all animals of the same area. We provide two types of algorithms for counting minimum-perimeter animals on two lattices, namely, the square and hexagonal lattices, and analyze and compare the algorithms.
The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation algorithm for strip-separable red-blue set cover problem with unit disks. Finally, we obtain a constant factor approximation algorithm for red-blue set cover problem with unit disks by combining our algorithm for the strip-separable problem with the results of Ambühl et al. [1]. Our methods involve a novel decomposition of the optimal solution to line-separable problem into blocks with special structure and extensions of the sweep-line technique of Erlebach and van Leeuwen [9].
Four distinct binary trees (left), and the two distinct plane trees associated to them (right), which are in turn identical as unordered trees
A binary tree (left) and (i) the six distinct binary trees, (ii) the five distinct plane trees and (iii) the four distinct unordered trees represented by its fringe subtrees (right)
A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in random trees under a generalized notion of distinctness, which allows for many different interpretations of what “distinct” trees are. The random tree models considered are simply generated trees and families of increasing trees (recursive trees, d -ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what ‘distinct’ means) in random trees with n vertices is $$n/\sqrt{\log n}$$ n / log n for simply generated trees and $$n/\log n$$ n / log n for increasing trees.
Hasse diagram of the boundedness relation between structural parameters explored in this paper and related ones. An edge between a parameter A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{A}$$\end{document} above and a parameter B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B}$$\end{document} below means that whenever A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{A}$$\end{document} is bounded for some graph class, then so is B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B}$$\end{document}. The parameters for which MESP is FPT are in green (dark if the result is described in this paper, light if implied by those described). Yellow represents a parameter for which MESP is FPT in combination with the desired eccentricity (again, dark if the result is described in this paper, light if implied by those described). The figure is inspired by [11]
Example of a situation from Lemma 11. Segment σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\sigma }$$\end{document} is highlighted in blue and D={v,v′}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = \{v, v'\}$$\end{document}
Example of a situation from Lemma 12. Path P is red, candidate segments are blue. The segment containing vertex u satisfies v. If all the dashed parts are present in G, then the segment containing vertex w also satisfies v
The Minimum Eccentricity Shortest Path Problem consists in finding a shortest path with minimum eccentricity in a given undirected graph. The problem is known to be NP-complete and W[2]-hard with respect to the desired eccentricity. We present fpt-algorithms for the problem parameterized by the modular width, distance to cluster graph, the combination of treewidth with the desired eccentricity, and maximum leaf number.
An illustration of the various result types as indicated in Table 1. Arrows indicate how a problem is generalized by another problem
Example of an antichain and its depth in a perfect 3-ary tree. We see that |pred(A)|=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\text {pred}(A)|=2$$\end{document}, but d(A)=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(A)=4$$\end{document}. If k=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document}, the dynamic programming algorithm will not compute S(A, t) since d(A)>k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(A)>k$$\end{document}. The only antichains with depth ≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le 2$$\end{document} are the empty set and the root node r on its own as a set. Indeed, d(∅)=d({r})=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(\emptyset ) = d(\{r\}) =1$$\end{document}. Note that for instances with k=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document}, a feasible schedule may exist. If so, we will find that R({r},1)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(\{r\},1)=1$$\end{document}, which will be defined later. In this way, we can still find the antichain A as a solution
Visualization of the definitions of M and B and the schedule σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*$$\end{document} in the proof of Lemma 6 is shown in a. b Depicts the schedule σ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma '$$\end{document} as chosen in the subcase d(B)>k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(B)>k$$\end{document}. The grey boxes indicate which jobs are processed in the schedules. We will prove that |D(A(σ′))|<|D(A(σ∗))|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|D(A(\sigma '))|<|D(A(\sigma ^*))|$$\end{document}
The fine-grained parameterized complexity of partial scheduling, where γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} denotes k-sched, Cmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\max }$$\end{document} and S.I. abbreviates Subgraph Isomorphism (Color table online)
We study a natural variant of scheduling that we call partial scheduling: in this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type f ( k ) n O ( 1 ) or n O ( f ( k ) ) exist for a function f that is as small as possible. Our contribution is two-fold: First, we categorize each variant to be either in P , N P -complete and fixed-parameter tractable by k, or W [ 1 ] -hard parameterized by k. Second, for many interesting cases we further investigate the runtime on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an O ( 8 k k ( | V | + | E | ) ) time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where G = ( V , E ) is the graph with precedence constraints.
Reduction from Hitting Set to Vertex Cover by ClsVD
It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called ‘structural parameterizations’, here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least kO(k)nO(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2O(k)nO(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in 2O(k)nO(1) time where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis.
The minimum hitting set of bundles problem (Mhsb) is a natural generalization of the minimum hitting set problem, where instead of hitting single elements, bundles of elements are hit. More specifically, we are given a ground set of elements and a family of sets. Every set in this family contains bundles of elements, which are subsets of the ground set. The task is to find a collection of elements of minimum size such that at least one bundle of every set in the family is hit. Motivated by several applications, we consider Mhsb restricted to interval and 2-dimensional interval bundles. We study the computational complexity and give polynomial-time algorithms for several classes of instances with these special structured bundles.
We study the problem of estimating the parameters of a Boolean product distribution in d dimensions, when the samples are truncated by a set S⊆{0,1}d accessible through a membership oracle. This is the first time that the computational and statistical complexity of learning from truncated samples is considered in a discrete setting. We introduce a natural notion of fatness of the truncation set S, under which truncated samples reveal enough information about the true distribution. We show that if the truncation set is sufficiently fat, samples from the true distribution can be generated from truncated samples. A stunning consequence is that virtually any statistical task (e.g., learning in total variation distance, parameter estimation, uniformity or identity testing) that can be performed efficiently for Boolean product distributions, can also be performed from truncated samples, with a small increase in sample complexity. We generalize our approach to ranking distributions over d alternatives, where we show how fatness implies efficient parameter estimation of Mallows models from truncated samples. Exploring the limits of learning discrete models from truncated samples, we identify three natural conditions that are necessary for efficient identifiability: (i) the truncation set S should be rich enough; (ii) S should be accessible through membership queries; and (iii) the truncation by S should leave enough randomness in all directions. By carefully adapting the Stochastic Gradient Descent approach of (Daskalakis et al., FOCS 2018), we show that these conditions are also sufficient for efficient learning of truncated Boolean product distributions.
Case 2(i)(d)
In an undirected graph G, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of G is the CFON chromatic number of G, denoted by χON(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)$$\end{document}. The decision problem that asks whether χON(G)≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le k$$\end{document} is NP-complete. Structural as well as algorithmic aspects of this problem have been well studied. We obtain the following results for χON(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)$$\end{document}:Bodlaender, Kolay and Pieterse (WADS 2019) showed the upper bound χON(G)≤fvs(G)+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le \mathsf{fvs}(G) + 3$$\end{document}, where fvs(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf{fvs}(G)$$\end{document} denotes the size of a minimum feedback vertex set of G. We show the improved bound of χON(G)≤fvs(G)+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le \mathsf{fvs}(G) + 2$$\end{document}, which is tight, thereby answering an open question in the above paper. We study the relation between χON(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)$$\end{document} and the pathwidth of the graph G, denoted pw(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf{pw}(G)$$\end{document}. The above paper from WADS 2019 showed the upper bound χON(G)≤2tw(G)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le 2\mathsf{tw}(G) + 1$$\end{document} where tw(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf{tw}(G)$$\end{document} stands for the treewidth of G. This implies an upper bound of χON(G)≤2pw(G)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le 2\mathsf{pw}(G) + 1$$\end{document}. We show an improved bound of χON(G)≤53(pw(G)+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le \left\lfloor \frac{5}{3}(\mathsf{pw}(G) + 1)\right\rfloor $$\end{document}. We prove new bounds for χON(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)$$\end{document} with respect to the structural parameters neighborhood diversity and distance to cluster, improving the existing results of Gargano and Rescigno (Theor. Comput. Sci. 2015) and Reddy (Theor. Comput. Sci. 2018), respectively. Furthermore, our techniques also yield improved bounds for the closed neighborhood variant of the problem. We prove bounds for Sk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k$$\end{document}-free graphs where Sk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k$$\end{document} is a star on k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document} vertices. For a graph G with maximum degree Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}, it is known that χON(G)≤Δ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)\le \Delta + 1$$\end{document} and this bound is tight in general. When G is Sk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k$$\end{document}-free, we show that χON(G)=O(k·log2+ϵΔ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)= O(k\cdot \log ^{2+\epsilon }\Delta )$$\end{document}, for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document}. In particular, when G is claw-free, this implies that χON(G)=O(log2+ϵΔ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{ON}(G)= O(\log ^{2 + \epsilon }\Delta )$$\end{document}. Further, we show existence of claw-free graphs that require Ω(logΔ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\log \Delta )$$\end{document} colors. We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let χON∗(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*_{ON}(G)$$\end{document} denote the minimum number of colors required to color G as per this variant. Abel et al. (SIDMA 2018) showed that χON∗(G)≤8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*_{ON}(G)\le 8$$\end{document} when G is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that χON∗(G)≤5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*_{ON}(G)\le 5$$\end{document} for all planar G. This approach also yields the bound χON∗(G)≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*_{ON}(G)\le 4$$\end{document} for all outerplanar G. All our bounds are a result of constructive algorithmic procedures.
Graph parameters and the complexity of Graph Burning. Our results in this paper are on the parameters with the dark background. Connections between two parameters imply the existence of a function in the one above (being in this sense more general) that lower-bounds the one below. “The maximum diameter of the components” is shortened as “max diameter”
The partition of the vertex set in the proof of Theorem 2.7
The reduction from Set Cover to Graph Burning. The edges in the cliques U2,⋯,Uk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{2}, \dots , U_{k-1}$$\end{document}, and U are omitted
Graph Burning asks, given a graph $$G = (V,E)$$ G = ( V , E ) and an integer k , whether there exists $$(b_{0},\dots ,b_{k-1}) \in V^{k}$$ ( b 0 , ⋯ , b k - 1 ) ∈ V k such that every vertex in G has distance at most i from some $$b_{i}$$ b i . This problem is known to be NP-complete even on connected caterpillars of maximum degree 3. We study the parameterized complexity of this problem and answer all questions by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless $$\mathrm {NP} \subseteq \mathrm {coNP/poly}$$ NP ⊆ coNP / poly . We also show that the problem is fixed-parameter tractable parameterized by clique-width plus the maximum diameter among all connected components. This implies the fixed-parameter tractability parameterized by modular-width, by treedepth, and by distance to cographs. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP.
D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1$$\end{document}, a forbidden subgraph for funnels
A digraph which is not a funnel. Removing the arcs (v, u) and (u, w) results in a funnel
Example application of shift neighbors (RR 3.4)
In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document}-Arc Deletion Set (C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document}-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document}. In this work, we choose C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel-ADS with O(k6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(k^6)$$\end{document} many vertices and O(k7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(k^7)$$\end{document} many arcs, computable in O(nm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(nm)$$\end{document} time, where n is the number of vertices and m the number of arcs in the input digraph.
Relations of our approximation results. An arc from problem A to B labeled f(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\alpha )$$\end{document} denotes the existence of an f(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\alpha )$$\end{document}-approximation for B, given an α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-approximation for A. In Corollary 16, α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} has to be constant. In Corollary 15, α(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (\cdot )$$\end{document} is a function of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. The ratio of 2-IM is by Theorem 8; combining this with Theorem 17 yields the ratio for 2-UM
Theorem 6: E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document} is curvy blue, E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document} is straight red. Marked vertices are encircled (Color figure online)
2-IM instance for k=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k{=}3$$\end{document} with an integrality gap ≥μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ge }\sqrt{\mu }$$\end{document}. E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document} is straight and red, E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document} is curvy and blue. Dotted green lines identify vertices (Color figure online)
2-IM instance G4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}} _4$$\end{document} as in Lemma 9. Edges in E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document} are curvy and blue, edges in E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document} straight and red. The vertices are labeled according to the first stage (Color figure online)
Visualization of the relationships between E∩,M∩∗,M1(i),Ri\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\cap , M^*_\cap , M^{(i)}_1, R_i$$\end{document} and Ri∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^*_i$$\end{document} for i∈[3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in [3]$$\end{document}
In multistage perfect matching problems, we are given a sequence of graphs on the same vertex set and are asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios. As our main contribution, we present the first non-trivial approximation algorithms for these problems: On the one hand, we devise a tight approximation on graph sequences of length two (2-stage graphs). On the other hand, we propose several general methods to deduce multistage approximations from blackbox approximations on 2-stage graphs.
Number of vertices (n) and minimum edge clique cover (cc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {cc}$$\end{document}) of PACE 2017 treewidth (TW) and minimum fill-in (MF) instances. Both values are truncated from above at 500 and the lines n=cc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = \texttt {cc}$$\end{document} and n/2=cc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/2 = \texttt {cc}$$\end{document} are shown
An example graph G with vertex set V(G)={a,b,c,d,e,f}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G) = \{a,b,c,d,e,f\}$$\end{document}. It holds that Ω={b,c,e}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega = \{b,c,e\}$$\end{document} is a PMC of G, and therefore by Proposition 2, {a}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a\}$$\end{document}, {d}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d\}$$\end{document}, and {f}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f\}$$\end{document} are blocks of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}. As an example of Proposition 3, consider Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} and the block C={a}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = \{a\}$$\end{document}. Now N(C)={b,c}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(C) = \{b,c\}$$\end{document} is a minimal separator contained in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}, and C′={d,e,f}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C' = \{d,e,f\}$$\end{document} is a full component of {b,c}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{b,c\}$$\end{document}. It holds that Ω⊆N[C′]={b,c,d,e,f}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subseteq N[C'] = \{b,c,d,e,f\}$$\end{document}
An example graph G with vertex set V(G)={x,y,z,xy,xz,yz,xyz}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G) = \{x,y,z,xy,xz,yz,xyz\}$$\end{document} and edge clique cover W={X,Y,Z}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}= \{X,Y,Z\}$$\end{document}, where X={x,xy,xz,xyz}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = \{x,xy,xz,xyz\}$$\end{document}, Y={y,xy,yz,xyz}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y = \{y,xy,yz,xyz\}$$\end{document}, and Z={z,xz,yz,xyz}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z = \{z,xz,yz,xyz\}$$\end{document}. Here, for example W[xy]={X,Y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}[xy] = \{X,Y\}$$\end{document}, W[{xy,xz}]={X,Y,Z}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}[\{xy,xz\}] = \{X,Y,Z\}$$\end{document}, V(G,{X,Y})={x,y,xy}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G,\{X,Y\}) = \{x,y,xy\}$$\end{document}, V(G,{X,Y},{Z})={x,y,xy,z}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G, \{X,Y\}, \{Z\}) = \{x,y,xy,z\}$$\end{document}. Also, C({X,Y})={{x,y,xy}}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(\{X,Y\}) = \{\{x,y,xy\}\}$$\end{document}, and because {x,y,xy}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x,y,xy\}$$\end{document} is a block, the part {X,Y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X,Y\}$$\end{document} is good. The parts {X}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X\}$$\end{document} and {Y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Y\}$$\end{document} are not compatible because V(G,{X},{Y})={x,y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G,\{X\},\{Y\}) = \{x,y\}$$\end{document} is not equal to V(G,{X,Y})={x,y,xy}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(G, \{X,Y\}) = \{x,y,xy\}$$\end{document}
We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ( $$\texttt {cc}$$ cc ) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $$\texttt {cc}$$ cc is at most the number of taxa, in fractional hypertreewidth $$\texttt {cc}$$ cc is at most the number of hyperedges, and in treewidth of Bayesian networks $$\texttt {cc}$$ cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most $$2^\texttt {cc}$$ 2 cc , the number of potential maximal cliques is at most $$3^\texttt {cc}$$ 3 cc , and these objects can be listed in times $$O^*(2^\texttt {cc})$$ O ∗ ( 2 cc ) and $$O^*(3^\texttt {cc})$$ O ∗ ( 3 cc ) , respectively, even when no edge clique cover is given as input; the $$O^*(\cdot )$$ O ∗ ( · ) notation omits factors polynomial in the input size. These enumeration algorithms imply $$O^*(3^\texttt {cc})$$ O ∗ ( 3 cc ) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give $$O^*(4^m)$$ O ∗ ( 4 m ) time and $$O^*(3^m)$$ O ∗ ( 3 m ) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size $$\texttt {cc}'$$ cc ′ is given as a part of the input we give $$O^*(2^{\texttt {cc}'})$$ O ∗ ( 2 cc ′ ) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an $$O^*(2^n)$$ O ∗ ( 2 n ) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities $$O^*(9^{\texttt {cc}'})$$ O ∗ ( 9 cc ′ ) and $$O^*(9^{\texttt {cc}+ O(\log ^2 \texttt {cc})})$$ O ∗ ( 9 cc + O ( log 2 cc ) ) for problems in this framework.
A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} and ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document}, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {NP}$$\end{document}-complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time O(9k·n9)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(9^k \cdot n^9)$$\end{document}, and also give a polynomial-time 9-approximation algorithm.
Construction of D for t=7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=7$$\end{document}
Construction of D for t=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1$$\end{document}
We initiate the parameterized complexity study of minimum t -spanner problems on directed graphs. For a positive integer t , a multiplicative t -spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G , that is, H keeps the distances in G up to the distortion (or stretch) factor t . An additive t -spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t , that is, the distances in H and G differ by at most t . The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k , decide whether G has a multiplicative t -spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t -spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k .
We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: For example, Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs . We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free (non-elastic) founder graph from a gapless , and (parameterized) near-linear time algorithms to construct a semi-repeat-free (repeat-free, respectively) elastic founder graph from general . Finally, we show that repeat-free founder graphs admit a reduction to Wheeler graphs in polynomial time.
This paper studies empty squares in arbitrary orientation among a set P of n points in the plane. We prove that the number of empty squares with four contact pairs is between Ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n)$$\end{document} and O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^2)$$\end{document}, and that these bounds are tight, provided P is in general position. A contact pair of a square is a pair of a point p∈P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in P$$\end{document} and a side ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} of the square with p∈ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in \ell $$\end{document}. The upper bound O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^2)$$\end{document} also applies to the number of empty squares with four contact points. Meanwhile, the lower bound becomes 0 as we can construct a point set among which there is no square of four contact points. These combinatorial results are based on new observations on the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty $$\end{document} Voronoi diagram with the axes rotated and its close connection to empty squares in arbitrary orientation. We then present an algorithm that maintains a combinatorial structure of the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty $$\end{document} Voronoi diagram of P, while the axes of the plane continuously rotate by 90 degrees, and simultaneously reports all empty squares with four contact pairs among P in an output-sensitive way within O(slogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(s\log n)$$\end{document} time and O(n) space, where s denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among P and a square annulus of minimum width or minimum area that encloses P over all orientations can be computed in worst-case O(n2logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^2 \log n)$$\end{document} time.
An example of a valid triple (Yi,Zi,Si)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Y_i, Z_i, S_i)$$\end{document}. A semicomplete digraph D, the set X=⋃i∈[q]Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=\bigcup _{i\in [q]}X_i$$\end{document} is such that mco(D-X)=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{mco}(D-X)=3$$\end{document} and C1,…,Cq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1,\ldots , C_q$$\end{document} are strong components of D-X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D-X$$\end{document}. Yi=C1′∪C2′∪⋯∪Ci′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i=C'_1\cup C'_2\cup \cdots \cup C'_i$$\end{document} and Zi=Ci+1′∪Ci+2′∪⋯∪Cq′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_i=C'_{i+1}\cup C'_{i+2}\cup \cdots \cup C'_q$$\end{document}, where Ci′=Ci∪Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C'_i=C_i\cup X_i$$\end{document}, i∈[q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in [q]$$\end{document}. The arcs uv, u∈Ci′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in C'_i$$\end{document}, v∈Cj′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in C'_j$$\end{document} for i<j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i<j$$\end{document} are omitted as well as the arcs within X between Xt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_t$$\end{document} and Ct\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_t$$\end{document}, t∈[q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [q]$$\end{document}. The set Si\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document} is the set of the three square vertices, one in each of Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document}, Xi+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i+1}$$\end{document}, and Xq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_q$$\end{document}. The set Si\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document} is a minimal vertex cover of the dashed arcs from Zi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_i$$\end{document} to Yi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document}. Note that the vertex in X1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1$$\end{document} is not in Si\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document} as the arc incident to it with the tail in Zi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_i$$\end{document} is already covered by Si\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document}. Note also the hollow circle vertex in Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document}, the only reason it is in X is to reduce the size of Ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_i$$\end{document} and as such it will not appear in any Sj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_j$$\end{document}, j∈[q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in [q]$$\end{document}, in the set of q valid triples defining these components
A directed graph D is semicomplete if for every pair x , y of vertices of D , there is at least one arc between x and y . Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph $$D=(V,A)$$ D = ( V , A ) and a pair of natural numbers k and $$\ell $$ ℓ , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in $$D-X$$ D - X has at most $$\ell $$ ℓ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $$\ell =1.$$ ℓ = 1 . We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: $$k, \ell ,\ell +k$$ k , ℓ , ℓ + k and $$n-\ell $$ n - ℓ . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time $$O^*(2^{16k})$$ O ∗ ( 2 16 k ) but not in time $$O^*(2^{o(k)})$$ O ∗ ( 2 o ( k ) ) unless the Exponential Time Hypothesis (ETH) fails. The upper bound $$O^*(2^{16k})$$ O ∗ ( 2 16 k ) implies the upper bound $$O^*(2^{16(n-\ell )})$$ O ∗ ( 2 16 ( n - ℓ ) ) for the parameter $$n-\ell .$$ n - ℓ . We complement the latter by showing that there is no algorithm of time complexity $$O^*(2^{o({n-\ell })})$$ O ∗ ( 2 o ( n - ℓ ) ) unless ETH fails. Finally, we improve (in dependency on $$\ell $$ ℓ ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $$\ell +k$$ ℓ + k on general digraphs from $$O^*(2^{O(k\ell \log (k\ell ))})$$ O ∗ ( 2 O ( k ℓ log ( k ℓ ) ) ) to $$O^*(2^{O(k\log (k\ell ))}).$$ O ∗ ( 2 O ( k log ( k ℓ ) ) ) . Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $$O^*(2^{o(k\log \ell )})$$ O ∗ ( 2 o ( k log ℓ ) ) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $$O^*(2^{o(k\log k)}).$$ O ∗ ( 2 o ( k log k ) ) .
The instance showing that no deterministic algorithm is better than m-competitive for Machine Covering. To the left, the best possible solution that an online algorithm can construct achieves minimum load 1. To the right, the optimal minimum load is m
A possible order Jσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {J}} ^\sigma $$\end{document} for an instance with k~<m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{k}} < m$$\end{document} large (dark) jobs. They partition the sequence into sets of small (light) jobs, each one having total size Si(Jσ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i({\mathcal {J}} ^\sigma )$$\end{document}, i=0,⋯,k~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=0,\dots ,{\tilde{k}}$$\end{document}
A comparison of the solution returned by the Greedy-strategy (left) and the optimal solution (right) for some order σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} of the instance defined in Theorem 4. Dark jobs have size 1 and the remaining jobs have size ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}
The classification of large (dark) and small (light and dashed) jobs. During sampling, all small jobs are misclassified (dashed ones). Threshold P↑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^\uparrow $$\end{document} classifies large jobs, while threshold τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} classifies small jobs. Jobs in between are conservatively classified as large, since misclassifying large jobs is fatal. Increasing τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} due to a small job is a helpful event as less small jobs will be misclassified. On the other hand, increasing τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} due to a large job below P↑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^\uparrow $$\end{document} is a fatal event as large jobs will be misclassified. The choice of P↑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^\uparrow $$\end{document} ensures that fatal events are unlikely to happen
In the Online Machine Covering problem, jobs, defined by their sizes, arrive one by one and have to be assigned to m parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. Unfortunately, the classical model allows only fairly pessimistic performance guarantees: The best possible deterministic ratio of m is achieved by the Greedy-strategy, and the best known randomized algorithm has competitive ratio O~(m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{O}}(\sqrt{m})$$\end{document}, which cannot be improved by more than a logarithmic factor. Modern results try to mitigate this by studying semi-online models, where additional information about the job sequence is revealed in advance or extra resources are provided to the online algorithm. In this work, we study the Machine Covering problem in the recently popular random-order model. Here, no extra resources are present but, instead, the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham’s Greedy-strategy in this context and establish that its competitive ratio decreases slightly to Θmlog(m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta \left( \frac{m}{\log (m)}\right) $$\end{document}, which is asymptotically tight. Then, as our main result, we present an improved O~(m4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{O}}(\root 4 \of {m})$$\end{document}-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound, showing that no algorithm can have a competitive ratio of Olog(m)loglog(m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left( \frac{\log (m)}{\log \log (m)}\right) $$\end{document} in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest.
Forbidden structures for reflexive interval digraphs (possibly b=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=c$$\end{document} in i, ii, iv and v). A dashed arc from u to v indicates the absence of the edge (u, v) in the graph. Note that the vertices are assumed to have self-loops since a vertex without a self-loop is itself forbidden in a reflexive interval digraph
An illustration of Theorem 1: the figure on the left shows a reflexive interval digraph with its vertex ordering (1, 2, 3, 4, 5) in which none of the structures in Fig. 1 occur. Using the ordering, as in the proof of Theorem 1, we can find an interval representation of the digraph. Following the proof, we get y1=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1=2$$\end{document}, y2=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2=5$$\end{document}, y3=y4=y5=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_3=y_4=y_5=6$$\end{document} and z1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_1=1$$\end{document}, z2=z3=z4=z5=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_2=z_3=z_4=z_5=0$$\end{document}. Also, y1′=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1'=3$$\end{document}, y2′=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2'=4$$\end{document}, y3′=y4′=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_3'=y_4'=5$$\end{document}, y5′=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_5'=6$$\end{document} and z1′=z2′=z3′=z4′=z5′=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_1'=z_2'=z_3'=z_4'=z_5'=0$$\end{document}. Then the collection of intervals, {S1=[1,1.17]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{S_1=[1,1.17]$$\end{document}, T1=[1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1=[1,2]$$\end{document}, S2=[2,4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_2=[2,4]$$\end{document}, T2=[2,3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2=[2,3]$$\end{document}, S3=[3,5]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_3=[3,5]$$\end{document}, T3=[1.17,4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_3=[1.17,4]$$\end{document}, S4=[4,5]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_4=[4,5]$$\end{document}, T4=[4,4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_4=[4,4]$$\end{document}, S5=[5,5]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_5=[5,5]$$\end{document}, T5=[5,5]}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_5=[5,5]\}$$\end{document} forms a reflexive interval representation of the given digraph
Example of a DUF-digraph that has no kernel
Inclusion relations between graph classes. In the diagram, there is an arrow from A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} to B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} if and only if the class B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} is contained in the class A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}. Moreover, each inclusion is strict. The problems studied are efficiently solvable in the classes shown in white, while they are NP-hard and/or APX-hard in the classes shown in gray (∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^*$$\end{document} the complexity of the Absorbing-Set problem on DUF-digraphs and reflexive DUF-digraphs remain open)
Given a digraph G, a set X⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subseteq V(G)$$\end{document} is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subseteq V(G)$$\end{document} is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality dominating set or absorbing set or kernel, and the problems of computing a maximum cardinality independent set or kernel, are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals (Su,Tu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_u,T_u)$$\end{document} can be assigned to each vertex u of G such that (u,v)∈E(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,v)\in E(G)$$\end{document} if and only if Su∩Tv≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_u\cap T_v\ne \emptyset $$\end{document}. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs—which arise when we require that the two intervals assigned to a vertex have to intersect. We see as our main contribution the identification of the class of reflexive interval digraphs as an important class of digraphs. We show that while the problems mentioned above are NP-complete, and even hard to approximate, on interval digraphs (even on some very restricted subclasses of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate), they are all efficiently solvable, in most of the cases linear-time solvable, in the class of reflexive interval digraphs. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs. In particular, we obtain a vertex ordering characterization of reflexive interval digraphs that implies the existence of an O(n+m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n+m)$$\end{document} time algorithm for computing a maximum cardinality independent set in a reflexive interval digraph, improving and generalizing the earlier known O(nm) time algorithm for the same problem for the interval nest digraphs. (Here m denotes the number of edges in the digraph not counting the self-loops.) We also show that reflexive interval digraphs are kernel-perfect and that a kernel in such digraphs can be computed in linear time. This generalizes and improves an earlier result that interval nest digraphs are kernel-perfect and that a kernel can be computed in such digraphs in O(nm) time. The structural characterizations that we show for point-point digraphs, apart from helping us construct the NP-completeness/APX-hardness reductions, imply that these digraphs can be recognized in linear time. We also obtain some new results for undirected graphs along the way: (a) We describe an O(n(n+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n(n+m))$$\end{document} time algorithm for computing a minimum cardinality (undirected) independent dominating set in cocomparability graphs, which slightly improves the existing O(n3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^3)$$\end{document} time algorithm for the same problem by Kratsch and Stewart; and (b) We show that the Red-Blue Dominating Set problem, which is NP-complete even for planar bipartite graphs, is linear-time solvable on interval bigraphs, which is a class of bipartite (undirected) graphs closely related to interval digraphs.
An example of the reduction from 3-Dimensional Matching to n-Multicolored Resolving Set in which U={1,2,3}×[n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=\{1,2,3\}\times [n]$$\end{document} and F={A1,...,Am}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{F}=\{A_1,...,A_m\}$$\end{document}. Here we only draw the corresponding paths and resolved pairs of the tuple Aj={(1,x),(2,y),(3,z)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_j=\{(1,x),(2,y),(3,z)\}$$\end{document}
An example showing a part of G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document}. Triangles represent corresponding forced vertex gadgets. Curves or lines represent paths. We use different colors, dotted and dashed lines just to avoid the chaos caused by many crowded curves or lines. For clarity, some forced vertex gadgets do not appear on the figure
The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph G. Here, a set S⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subseteq V(G)$$\end{document} is resolving if no two distinct vertices of G have the same distance vector to S. The complexity of Metric Dimension in graphs of bounded treewidth remained elusive in the past years. Recently, Bonnet and Purohit [IPEC 2019] showed that the problem is W[1]-hard under treewidth parameterization. In this work, we strengthen their lower bound to show that Metric Dimension is NP-hard in graphs of treewidth 24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$24$$\end{document}.
The binomial tree T4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_4$$\end{document}, where the labels denote the color of each vertex in a first-fit coloring achieving the highest possible color (Color figure online)
The barriers in a propagation gadget for Grundy Coloring. The biclique has small Grundy number but do not propagate, nor it imposes a unique choice. The three other propagations have arbitrary large Grundy number, rendering them useless for a parameterized reduction
The encoding Hi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_i$$\end{document} of one Vi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_i$$\end{document} ordered u1<u2<u3<u4<u5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_1< u_2< u_3< u_4 < u_5$$\end{document}. In bold, a possible independent set intersecting the five sets and containing a consistent pair l(u), r(u)
The biclique Kt,q-9(i,j)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{t,q-9}(i,j)$$\end{document} encoding the pairs Pi,j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{i,j}$$\end{document}, and its connection to the two neighboring horizontal half-graphs, with n=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=5$$\end{document}, t=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=10$$\end{document}, and q=14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=14$$\end{document}
The rounded rectangles represent the Bi,j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{i,j}$$\end{document}, and the numbers therein, the z∈[18]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in [18]$$\end{document} such that dz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_z$$\end{document} is fully linked to it. These are the “colors” that a center in Ai,j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i,j}$$\end{document} will have to fight for. The circles represent the half-graphs, and the number therein, the only z such that dz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_z$$\end{document} is not linked to it. Red integers are offset by 9 (, , and so on). The edges represent the non-empty interaction between the Ai,j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i,j}$$\end{document} and the half-graphs. The structure is glued like a torus (Color figure online)
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n2k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(k)n^{2^{k-1}}$$\end{document}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document}-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS ’17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on Kt,t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{t,t}$$\end{document}-free graphs for b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document}-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity Rossman (SIAM J. Comput. 43:256–279, 2014), DNF sparsification Gopalan et al. (Comput. Complex. 22:275–310 2013), randomness extractors Li et al. (In: APPROX-RANDOM, LIPIcs 116:51:1–13, 2018), and recent advances on the Erdős-Rado sunflower conjecture Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) Lovett et al. (From dnf compression to sunflower theorems via regularity, 2019) Rao (Discrete Anal. 8,2020). The recent breakthrough of Alweiss, Lovett, Wu and Zhang Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) gives an improved bound on the maximum size of a w -set system that excludes a robust sunflower. In this paper, we use this result to obtain an $$\exp (n^{1/2-o(1)})$$ exp ( n 1 / 2 - o ( 1 ) ) lower bound on the monotone circuit size of an explicit n -variate monotone function, improving the previous best known $$\exp (n^{1/3-o(1)})$$ exp ( n 1 / 3 - o ( 1 ) ) due to Andreev (Algebra and Logic, 26:1–18, 1987) and Harnik and Raz (In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, New York, 2000). We also show an $$\exp (\varOmega (n))$$ exp ( Ω ( n ) ) lower bound on the monotone arithmetic circuit size of a related polynomial via a very simple proof. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an $$n^{\varOmega (k)}$$ n Ω ( k ) lower bound on the monotone circuit size of the CLIQUE function for all $$k \leqslant n^{1/3-o(1)}$$ k ⩽ n 1 / 3 - o ( 1 ) , strengthening the bound of Alon and Boppana (Combinatorica, 7:1–22, 1987).
The tuple T=⟨{x2,x3,x4};N(w1),N(w2),N(w3),∅,∅⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}= \langle \{x_2, x_3, x_4\}; N(w_1), N(w_2), N(w_3), \emptyset , \emptyset \rangle $$\end{document} is a type. The matching {x2w1,x3w2,x4w3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_2w_1, x_3w_2, x_4w_3\}$$\end{document} is of T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}$$\end{document}
Inverting components and its symbolic representation used in the reduction from Red Blue Dominating Set to Maximum Edge Colorable Subgraph
(Left Figure) A red-gadget made for a vertex of degree four. The gadget is made from ten modules and have four output pairs of edges. More generally, it can be made from 2(d+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(d + 1)$$\end{document} modules and has d output pairs of edges. In the modified red-gadget, vertex r is replaced by two vertices r1,r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_1, r_2$$\end{document} as shown in left-top corner. (Right Figure) A blue-gadget made for a vertex of degree three. More generally, it can be made from d modules and has d input pairs of edges
A graph H is p-edge colorable if there is a coloring ψ:E(H)→{1,2,⋯,p}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : E(H) \rightarrow \{1,2,\dots ,p\}$$\end{document}, such that for distinct uv,vw∈E(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$uv, vw \in E(H)$$\end{document}, we have ψ(uv)≠ψ(vw)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (uv) \ne \psi (vw)$$\end{document}. The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that |E(H)|≥l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|E(H)| \ge l$$\end{document}. We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p+k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+k$$\end{document}, where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) l-mm(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l - {\texttt {mm}}(G)$$\end{document}, where mm(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\texttt {mm}}(G)$$\end{document} is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with O(k·p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(k \cdot p)$$\end{document} vertices. Furthermore, we show that there is no kernel of size O(k1-ϵ·f(p))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(k^{1-\epsilon } \cdot f(p))$$\end{document}, for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document} and computable function f, unless NP⊆coNP/poly\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf {NP}}\subseteq \textsf {coNP/poly}$$\end{document}.
Optimal adaptive strategy for the unanimous vote function, if x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} is the first test
Consider the following Stochastic Score Classification problem. A doctor is assessing a patient’s risk of developing a disease and can perform n different binary tests on the patient. The probability that test i is positive is pi and the outcomes of the n tests are independent. A patient’s score is the total number of positive tests. Possible scores thus range between 0 and n. This range is divided into subranges, corresponding to risk classes (e.g., LOW, MEDIUM, or HIGH risk). Each test has an associated cost. To reduce testing cost, instead of performing all tests and determining an exact score, the doctor can perform tests sequentially and stop testing when it is possible to determine the patient’s risk class. The problem is to determine the order in which the doctor should perform the tests, so as to minimize expected testing cost. We address the unit-cost case of the Stochastic Score Classification problem, and provide polynomial-time approximation algorithms for adaptive and non-adaptive versions of the problem. We also pose a number of open questions.
Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching , longest common substring , matching statistics , and longest repeated substring . Recent lines of research focused on extending some of these problems to vertex-labeled graphs, either by using efficient ad-hoc approaches which do not generalize to all input graphs, or by indexing difficult graphs and having worst-case exponential complexities. In the absence of an ubiquitous and polynomial tool like the suffix tree for labeled graphs, we introduce the labeled direct product of two graphs as a general tool for obtaining optimal algorithms in the worst case: we obtain conceptually simpler algorithms for the quadratic problems of string matching () and longest common substring () in labeled graphs. Our algorithms run in time linear in the size of the labeled product graph, which may be smaller than quadratic for some inputs, and their run-time is predictable, because the size of the labeled direct product graph can be precomputed efficiently. We also solve on graphs containing cycles, which was left as an open problem by Shimohira et al. in 2011. To show the power of the labeled product graph, we also apply it to solve the matching statistics () and the longest repeated string () problems in labeled graphs. Moreover, we show that our (worst-case quadratic) algorithms are also optimal, conditioned on the Orthogonal Vectors Hypothesis. Finally, we complete the complexity picture around by studying it on undirected graphs.
In this paper, we study the following pattern search problem: Given a pair of point sets A and B in fixed dimensional space Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}, with |B|=n,|A|=m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|B| = n,|A| = m$$\end{document} and n≥m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge m$$\end{document}, the pattern search problem is to find the translations T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document}’s of A such that each of the identified translations induces a matching between T(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(A)$$\end{document} and a subset B′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'$$\end{document} of B with cost no more than some given threshold, where the cost is defined as the minimum bipartite matching cost of T(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(A)$$\end{document} and B′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'$$\end{document}. We present a novel algorithm to produce a small set of candidate translations for the pattern search problem. For any B′⊆B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B' \subseteq B$$\end{document} with |B′|=|A|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|B'| = |A|$$\end{document}, there exists at least one translation T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document} in the candidate set such that the minimum bipartite matching cost between T(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(A)$$\end{document} and B′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'$$\end{document} is no larger than (1+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\epsilon )$$\end{document} times the minimum bipartite matching cost between A and B′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'$$\end{document} under any translation (i.e., the optimal translational matching cost). We also show that there exists an alternative solution to this problem, which constructs a candidate set of size Od,ϵ(nlog2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_{d,\epsilon }(n \log ^2 n)$$\end{document} in Od,ϵ(nlog2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_{d,\epsilon }(n \log ^2 n)$$\end{document} time with high probability of success. As a by-product of our construction, we obtain a weak ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}-net for hypercube ranges, which significantly improves the construction time and the size of the candidate set. Our technique can be applied to a number of applications, including the translational pattern matching problem.
The parse tree of some 3-expression of P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4$$\end{document}
Transformation of Lemma 8
Recently, independent groups of researchers have presented algorithms to compute a maximum matching in O~(f(k)·(n+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal{O}}(f(k) \cdot (n+m))$$\end{document} time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4$$\end{document}-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle’s theorem to this problem directly, because a matching cannot be expressed in MSO1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$MSO_1$$\end{document} logic. Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. We also present how to compute the Edmonds-Gallai decomposition in almost linear time by using the same framework. For that, we do apply Courcelle’s theorem but to the classic Tutte-Berge formula, that can easily be expressed as a CMSO1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CMSO_1$$\end{document} optimization problem. Doing so, we can compute the cardinality of a maximum matching, but not the matching itself. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs and with a distributed version of Courcelle’s theorem (Courcelle and Vanicat, DAM 2016) – of which we present here a slightly stronger version than the standard one in the literature. Finally, for the bipartite graphs of clique-width at most k, we present an alternative O~(k2·(n+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal{O}}(k^2\cdot (n+m))$$\end{document}-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC’21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.
A diamond, a gem, a house, a domino, a bull, and a dart
Two cases for g1,g2,g3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_1, g_2, g_3$$\end{document}. The left figure shows the case when they all meet at the same vertex w. The right figure shows when g1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_1$$\end{document} and g2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_2$$\end{document} meet first at u and meet g3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_3$$\end{document} with w later. Real lines indicate an individual arc and dotted lines indicate a directed path
For a family of graphs F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document}, WeightedF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document}-Deletion is the problem for which the input is a vertex weighted graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V, E)$$\end{document} and the goal is to delete S⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subseteq V$$\end{document} with minimum weight such that G\S∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \setminus S \in \mathcal {F}$$\end{document}. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.
Nodes u1∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_1^*$$\end{document} and u2∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_2^*$$\end{document} are ancestors of u1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_1$$\end{document} and u2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_2$$\end{document} respectively. They are induced since their subtrees have leaf label 3 in common. Note that u1∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_1^*$$\end{document} and t are also induced but we don’t report them as the answer to the HIA problem since they have a lower combined weight (11+10 = 21) compared to W(u1∗)+W(u2∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(u_1^*) + W(u_2^*)$$\end{document} which is 24
We refer to Sect. 3.2 for the description of this figure
Let T1 and T2 be two rooted trees with an equal number of leaves. The leaves are labeled, and the labeling of the leaves in T2 is a permutation of those in T1. Nodes are associated with weight, such that the weight of a node u, denoted by W(u), is more than the weight of its parent. A node x∈T1 and a node y∈T2 are induced, iff their subtrees have at least one common leaf label. A heaviest induced ancestor query HIA(u1,u2) with input nodes u1∈T1 and u2∈T2 asks to output the pair (u1∗,u2∗) of induced nodes with the highest combined weight W(u1∗)+W(u2∗), such that u1∗ is an ancestor of u1 and u2∗ is an ancestor of u2. This is a useful primitive in several text processing applications. Gagie et al. (Proceedings of the 25th Canadian Conference on Computational Geometry, CCCG 2013, Waterloo, Ontario, Canada, 2013) introduced this problem and proposed three data structures with the following space-time trade-offs: (i) O(nlog2n) space and O(lognloglogn) query time, (ii) O(nlogn) space and O(log2n) query time, and (iii) O(n) space and O(log3+ϵn) query time. Here n is the number of nodes in both trees combined and ϵ>0 is an arbitrarily small constant. We present two new data structures with better space-time trade-offs: (i) O(nlogn) space and O(lognloglogn) query time, and (ii) O(n) space and O(log2n/loglogn) query time. Additionally, we present new applications of these results.
We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of $$O(5^k\cdot n\cdot m)$$ O ( 5 k · n · m ) . We also present the concept of temporal distance , which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in $$O((8k)^d5^ k\cdot k\cdot n\cdot m)$$ O ( ( 8 k ) d 5 k · k · n · m ) time. Lastly, we introduce an $$O(6^kk!\cdot k\cdot n^2)$$ O ( 6 k k ! · k · n 2 ) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.
T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T'$$\end{document} is a partially grown DFS tree. c1,c2,c3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1,c_2,c_3$$\end{document} are the connected components in the subgraph induced by the unvisited vertices. For the sake of clarity, edges incident on c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2$$\end{document} and c3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_3$$\end{document} from the T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T'$$\end{document} are omitted
a A DFS tree T. b Heavy light decomposition of T and the resulting paths in P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}$$\end{document}. c The corresponding shallow tree
Let G be an undirected graph. We address the problem of fault tolerant depth first search (DFS) tree defined as follows. Build a compact data structure that, given any set F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}$$\end{document} of failed vertices or edges, can efficiently report a DFS tree of G\F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G{\backslash } {{\mathcal {F}}}$$\end{document}. We present an algorithm which is drastically simpler and yet more efficient than the current state-of-the-art algorithms for this problem. Additionally, for achieving efficiency, the current-state-of-the-algorithms have to crucially rely on sophisticated data structures. The simplicity of our algorithm also enables us to replace these sophisticated data structures with much simpler and lighter data structures that occupy optimal space and take optimal preprocessing time. Our algorithm for the fault tolerant DFS tree also leads to a better time complexity for maintaining a DFS tree in a fully dynamic environment.
For some β≥1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ge 1/2$$\end{document}, a Δβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta }$$\end{document}-metric graph G=(V,E,w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E,w)$$\end{document} is a complete edge-weighted graph such that w(v,v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(v,v)=0$$\end{document}, w(u,v)=w(v,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(u,v)=w(v,u)$$\end{document}, and w(u,v)≤β·(w(u,x)+w(x,v))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(u,v) \le \beta \cdot (w(u,x)+w(x,v))$$\end{document} for all vertices u,v,x∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u,v,x\in V$$\end{document}. A graph H=(V′,E′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=(V', E')$$\end{document} is called a spanning subgraph of G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} if V′=V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V'=V$$\end{document} and E′⊆E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E'\subseteq E$$\end{document}. Given a positive integer p, let H be a spanning subgraph of G satisfying the three conditions: (i) there exists a vertex subset C⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subseteq V$$\end{document} such that C forms a clique of size p in H; (ii) the set V\C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \setminus C$$\end{document} forms an independent set in H; and (iii) each vertex v∈V\C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in V \setminus C$$\end{document} is adjacent to exactly one vertex in C. The vertices in C are called hubs and the vertices in V\C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\setminus C$$\end{document} are called non-hubs. The Δβ-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta }\text {-}p$$\end{document}-Hub Center Problem (Δβ-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta }\text {-}p$$\end{document}HCP) is to find a spanning subgraph H of G satisfying all the three conditions such that the diameter of H is minimized. In this paper, we study Δβ-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta } \text {-} p$$\end{document}HCP for all β≥12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ge \frac{1}{2}$$\end{document}. We show that for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document}, to approximate Δβ-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta }\text {-}p$$\end{document}HCP to a ratio g(β)-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )-\epsilon $$\end{document} is NP-hard and we give r(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\beta )$$\end{document}-approximation algorithms for the same problem where g(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )$$\end{document} and r(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\beta )$$\end{document} are functions of β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. For 3-32<β≤5+510\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3-\sqrt{3}}{2}<\beta \le \frac{5+\sqrt{5}}{10}$$\end{document}, we give an approximation algorithm that reaches the lower bound of approximation ratio g(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )$$\end{document} where g(β)=3β-2β23(1-β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )= \frac{3\beta -2\beta ^2}{3(1-\beta )}$$\end{document} if 3-32<β≤23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}$$\end{document} and g(β)=β+β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta ) = \beta +\beta ^2$$\end{document} if 23≤β≤5+510\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2}{3}\le \beta \le \frac{5+\sqrt{5}}{10}$$\end{document}. For 5+510≤β≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{5+\sqrt{5}}{10}\le \beta \le 1$$\end{document}, we show that g(β)=4β2+3β-15β-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta ) =\frac{4\beta ^2+3\beta -1}{5\beta -1}$$\end{document} and r(β)=min{β+β2,4β2+5β+15β+1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\beta )= \min \{\beta +\beta ^2, \frac{4\beta ^2+5\beta +1}{5\beta +1}\}$$\end{document}. Additionally, for β≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ge 1$$\end{document}, we show that g(β)=β·4β-13β-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta ) = \beta \cdot \frac{4\beta -1}{3\beta -1}$$\end{document} and r(β)=min{β2+4β3,2β}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\beta )=\min \{\frac{\beta ^2+4\beta }{3},2\beta \}$$\end{document}. For β≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ge 2$$\end{document}, the upper bound on the approximation ratio r(β)=2β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\beta ) =2\beta $$\end{document} is linear in β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. For 3-32<β≤5+510\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3-\sqrt{3}}{2}<\beta \le \frac{5+\sqrt{5}}{10}$$\end{document}, we give an approximation algorithm that reaches the lower bound of approximation ratio g(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )$$\end{document} where g(β)=3β-2β23(1-β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )= \frac{3\beta -2\beta ^2}{3(1-\beta )}$$\end{document} if 3-32<β≤23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}$$\end{document} and g(β)=β+β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta ) = \beta +\beta ^2$$\end{document} if 23≤β≤5+510\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2}{3}\le \beta \le \frac{5+\sqrt{5}}{10}$$\end{document}. For β≤3-32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \le \frac{3 - \sqrt{3}}{2}$$\end{document}, we show that g(β)=r(β)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\beta )=r(\beta )=1$$\end{document}, i.e.,Δβ-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\beta }\text {-} p$$\end{document}HCP is polynomial-time solvable.
The dispersion of a point set $$P\subset [0,1]^d$$ P ⊂ [ 0 , 1 ] d is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect P . It was observed only recently that, for any $$\varepsilon >0$$ ε > 0 , certain randomized constructions provide point sets with dispersion smaller than $$\varepsilon $$ ε and number of elements growing only logarithmically in d . Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in d . Note that, however, the running-time will be super-exponential in $$\varepsilon ^{-1}$$ ε - 1 . Our construction is based on the apparently new insight that low-dispersion point sets can be deduced from solutions of certain k -restriction problems, which are well-known in coding theory.
Given a set of players P, a set of indivisible resources R, and a set of non-negative values {vpr}p∈P,r∈R, an allocation is a partition of R into disjoint subsets {Cp}p∈P so that each player p is assigned the resources in Cp. The max-min fair allocation problem is to determine the allocation that maximizes minp∑r∈Cpvpr. In the restricted case of this problem, each resource r has an intrinsic value vr, and vpr=vr for every player p who desires r and vpr=0 for every player p who does not. We study the restricted max-min fair allocation problem in this paper. For this problem, the configuration LP has played an important role in estimating and approximating the optimal solution. Our first result is an upper bound of 32126 on the integrality gap, which is currently the best. It is obtained by a tighter analysis of the local search of Asadpour et al. [TALG’12]. It remains unknown whether this local search runs in polynomial time or not. Our second result is a polynomial-time algorithm that achieves an approximation ratio of 4+δ for any constant δ∈(0,1). Our algorithm can be seen as a generalization of the aforementioned local search.
The quartets ab||cd, ac||bd, and ad||bc represent the first, second, and third phylogenetic trees for a, b, c, d from the left, respectively. ad|bc, for example, represents one of the two phylogenetic trees in the dotted curve, that is, ad||bc or the star graph with leaves a, b, c, d
An example of phylogenetic tree T for {a,b,c,d,e,f,g,h,i,j}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a,b,c,d,e,f,g,h,i, j\}$$\end{document}. T displays, for example, ac||gi, ac|gi, and ac|df, ad|cf, af|cd. The quartet trees ac|df, ad|cf, af|cd follow from the same substructure of T, the star formed by the restriction of T to a, c, d, f
The outline of our algorithm for reconstructing a phylogenetic tree from a complete A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}-partite system Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Q}}$$\end{document} defined in Example 2.1
A phylogenetic tree is a graphical representation of an evolutionary history of taxa in which the leaves correspond to the taxa and the non-leaves correspond to speciations. One of important problems in phylogenetic analysis is to assemble a global phylogenetic tree from small phylogenetic trees, particularly, quartet trees. Quartet Compatibility is the problem of deciding whether there is a phylogenetic tree inducing a given collection of quartet trees, and to construct such a phylogenetic tree if it exists. It is known that Quartet Compatibility is NP-hard and that there are only a few results known for polynomial-time solvable subclasses. In this paper, we introduce two novel classes of quartet systems, called complete multipartite quartet system and full multipartite quartet system, and present polynomial-time algorithms for Quartet Compatibility for these systems.
The decomposition tree, T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T'$$\end{document}, and the decompositions ∪{Tx}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cup \{T_{x}\}$$\end{document}
The tree TM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{M}$$\end{document}
Hliněný’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hliněný’s Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hliněný’s Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.
In this paper we present linear time approximation schemes for several generalized matching problems on nonbipartite graphs. Our results include $O_\epsilon(m)$-time algorithms for $(1-\epsilon)$-maximum weight $f$-factor and $(1+\epsilon)$-approximate minimum weight $f$-edge cover. As a byproduct, we also obtain direct algorithms for the exact cardinality versions of these problems running in $O(m\sqrt{f(V)})$ time. The technical contributions of this work include an efficient method for maintaining {\em relaxed complementary slackness} in generalized matching problems and approximation-preserving reductions between the $f$-factor and $f$-edge cover problems.
Possible adversarial graph with 6 vertices against example algorithm, having the red vertices as the accepted ones at the end (Color figure online)
Illustration of the subgraph discussed in Lemma 2
Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2.
Compact hash tables store a set S of n key-value pairs, where the keys are from the universe U={0,…,u-1}, and the values are v-bit integers, in close to B(u,n)+nv bits of space, where B(u,n)=log2un is the information-theoretic lower bound for representing the set of keys in S, and support operations insert, delete and lookup on S. Compact hash tables have received significant attention in recent years, and approaches dating back to Cleary [IEEE T. Comput, 1984], as well as more recent ones have been implemented and used in a number of applications. However, the wins on space usage of these approaches are outweighed by their slowness relative to conventional hash tables. In this paper, we demonstrate that compact hash tables based upon a simple idea of bucketing practically outperform existing compact hash table implementations in terms of memory usage and construction time, and existing fast hash table implementations in terms of memory usage (and sometimes also in terms of construction time), while having competitive query times. A related notion is that of a compact hash ID map, which stores a set S^ of n keys from U, and implicitly associates each key in S^ with a unique value (its ID), chosen by the data structure itself, which is an integer of magnitude O(n), and supports inserts and lookups on S^, while using space close to B(u,n) bits. One of our approaches is suitable for use as a compact hash ID map.
Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges $$T\subset \mathcal {P}(V)$$ T ⊂ P ( V ) , whose elements have cardinality three each, a null labelling is an assignment of $$\pm 1$$ ± 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled $$+1$$ + 1 and $$-1$$ - 1 . A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282–294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.
An edge {y,z}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{y,z\}$$\end{document} is cut by the ball BT(x,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_T(x,b)$$\end{document} if dT(x,y)<b<dT(x,z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_T(x,y)<b<d_T(x,z)$$\end{document}. The edge that being cut are depicted in orange (on the left). To subdivide an edge e={x,y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=\{x,y\}$$\end{document} of weight w the following steps are taken: (1) Delete the edge e. (2) Add a new vertex ve\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_e$$\end{document}. (3) Add two new edges {x,ve},{ve,y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x,v_e\},\{v_e,y\}$$\end{document} with weights α·w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \cdot w$$\end{document} and (1-α)·w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\alpha )\cdot w$$\end{document} for some α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}. On the right we subdivide the edges, so that no edge is cut (Color figure online)
Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any $n$-point Euclidean metric has an $O(t)$-spanner with $\tilde{O}(n^{1+1/t^2})$ edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of $\ell_p$ with $1<p\le 2$. Second, our construction yields a spanner which is both sparse and also {\em light}, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any $n$-point subset of $\ell_p$ for $1<p\le 2$ has an $O(t)$-spanner with $n^{1+\tilde{O}(1/t^p)}$ edges and lightness $n^{\tilde{O}(1/t^p)}$. In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an $O(t)$-spanner with lightness $O(n^{1/t})$. We exhibit the following tradeoff: metrics with decomposability parameter $\nu=\nu(t)$ admit an $O(t)$-spanner with lightness $\tilde{O}(\nu^{1/t})$. For example, $n$-point Euclidean metrics have $\nu\le n^{1/t}$, metrics with doubling constant $\lambda$ have $\nu\le\lambda$, and graphs of genus $g$ have $\nu\le g$. While these families do admit a ($1+\epsilon$)-spanner, its lightness depend exponentially on the dimension (resp. $\log g$). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.
We provide algorithmic methods for connected reconfiguration of lattice-based cellular structures by finite-state robots, motivated by large-scale constructions in space. We present algorithms that are able to detect and reconfigure arbitrary polyominoes, while also preserving connectivity of a structure during reconfiguration; we also provide mathematical proofs and performance guarantees. Specific results include methods for determining a bounding box, scaling a given arrangement, and adapting more general algorithms for transforming polyominoes.
Top-cited authors
Cecilia R. Aragon
  • University of Washington Seattle
Mark Manasse
Larry Rudolph
  • Two Sigma Investments
David Ron Karger
  • Massachusetts Institute of Technology
Giuseppe Liotta
  • Università degli Studi di Perugia