[Show abstract][Hide abstract] ABSTRACT: We study complexity issues related to some coloring problems in grids: we examine in particular the case of List coloring, of Precoloring extension and of (p,q)(p,q)-List coloring, the case of List coloring in bipartite graphs where lists in the first part of the bipartition are all of size pp and lists in the second part are of size qq. In particular, we characterize the complexity of (p,q)(p,q)-List coloring in grid graphs, showing that the only NP-complete case is (2, 3)-List coloring with k≥4k≥4 colors. We also show that Precoloring extension with 3 colors is NP-complete in subgrids.
[Show abstract][Hide abstract] ABSTRACT: Let G=(V,E) be a graph in which every vertex v∈V has a weight w(v)⩾0 and a cost c(v)⩾0. Let SG be the family of all maximum-weight stable sets in G. For any integer d⩾0, a minimum d-transversal in the graph G with respect to SG is a subset of vertices T⊆V of minimum total cost such that |T∩S|⩾d for every S∈SG. In this paper, we present a polynomial-time algorithm to determine minimum d-transversals in bipartite graphs. Our algorithm is based on a characterization of maximum-weight stable sets in bipartite graphs. We also derive results on minimum d-transversals of minimum-weight vertex covers in weighted bipartite graphs.
Journal of Discrete Algorithms 12/2012; 17:95-102.
[Show abstract][Hide abstract] ABSTRACT: In threshold graphs one may find weights for the vertices and a threshold value t such that for any subset S of vertices, the sum of the weights is at most the threshold t if and only if the set S is a stable (independent) set. In this note we ask a similar question about vertex colorings: given an integer p, when is it possible to find weights (in general depending on p) for the vertices and a threshold value tp such that for any subset S of vertices the sum of the weights is at most tp if and only if S generates a subgraph with chromatic number at most p−1? We show that threshold graphs do have this property and we show that one can even find weights which are valid for all values of p simultaneously.
[Show abstract][Hide abstract] ABSTRACT: A general formulation of the problems we are going to consider may be sketched as follows: we are given a system S, which is operated by an actor A; this actor tries to choose among several optimal actions that may be represented by subsets of S. An opponent O wants to prevent actor A from operating S in an optimum way by destroying some part P of S. O may, in particular, wish to find a part P of S as small as possible whose removal will reduce the efficiency of the operation of the system S by a given amount. Another way for O would be to determine a smallest possible part P (the most vital elements), which hits in a sufficient amount every possible optimal action of A. The goal of this chapter is to give a partial survey of such situations while focusing on simple models based on graphs and other (hopefully tractable) combinatorial structures. We have deliberately decided not to include proofs whenever they could be found in the original papers; for a few results that have not yet appeared elsewhere, proofs are given in extenso.
Progress in Combinatorial Optimization, 11/2011: pages 203-222; ISTE-WILEY., ISBN: 978-1-84821-206-0
[Show abstract][Hide abstract] ABSTRACT: Given an integer d and a weighted tree T, we show how to find in polynomial time a minimum d-transversal of all maximum-weight stable sets in T, i.e., a set of vertices of minimum size having at least d vertices in common with every maximum-weight stable set. Our proof relies on new structural results for maximum-weight stable sets on trees.
European Conference on Combinatorics, Graph Theory and Applications; 10/2011
[Show abstract][Hide abstract] ABSTRACT: We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ℘. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings,
s–t paths and s–t cuts in graphs) and we study d-transversals and d-blockers of stable sets or vertex covers in bipartite and in split graphs.
Journal of Combinatorial Optimization 01/2011; 22:857-872. · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.
[Show abstract][Hide abstract] ABSTRACT: In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases.We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied to derive some of these results.
Annals of Operations Research 06/2010; 175(1):287-307. · 1.03 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given an undirected graph G=(V,E) with matching number �nu(G), a d-blocker is a subset of edges B such thatnu(V,E\B)\leq nu(G)-d and a d-transversal T is a subset of edges such that every maximum matching M has |M\cap T|\geq d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree.
[Show abstract][Hide abstract] ABSTRACT: We study a multiprocessor extension of the preemptive open shop scheduling problem, where the set of processors is partitioned into processor groups. We show that the makespan minimization problem is polynomially solvable for two multiprocessor groups even if preemptions are restricted to integral times.
Operations Research Letters 01/2010; 38:129-132. · 0.52 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph G a k-coloring, i.e., a partition V1,..., Vk of the vertex set of G such that, for some specified neighborhood N(v) of each vertex v, the number of vertices in N(v)\cap Vi is (at most) a given integer h_i^v . The complexity of some variations is discussed according to NQ .v/, which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when G is a special tree).
[Show abstract][Hide abstract] ABSTRACT: A magnet is a pair u, v of adjacent vertices such that the proper neighbours of u are completely linked to the proper neighbours of v. It has been shown that one can reduce the graph by removing the two vertices u, v of a magnet and introducing a new vertex linked to all common neighbours of u and v without changing the stability number. We prove that all graphs containing no chordless cycle C-k (k >= 5) and none of eleven forbidden subgraphs can be reduced to a stable set by repeated use of magnets. For such graphs a polynomial algorithm is given to determine the stability number.
Graphs and Combinatorics 11/2009; · 0.35 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given an undirected graph G=(V,E) with matching number �nu(G), we define d-blockers as subsets of edges B such that nu((V,E\B))\leq nu(G)-d. We define d-transversals T as subsets of edges such that every maximum matching M has |M\cap T|\geq d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, chains and cycles and we construct minimum d-transversals and d-blockers in these special graphs. We also
study the complexity status of finding minimum transversals and blockers in arbitrary graphs.
[Show abstract][Hide abstract] ABSTRACT: We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P8-free bipartite graphs, but polynomial for P5-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme. oui