[Show abstract][Hide abstract] ABSTRACT: In this paper we design {\sf FPT}-algorithms for two parameterized problems.
The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$
and a list of allowed vertices of $H$ for every vertex of $G$, the question is
whether there exists a homomorphism from $G$ to $H$ respecting the list
constraints. The second problem is a variant of \textsc{Multiway Cut}, namely
\textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer
$\ell$, and a set $T$ of $r$ terminals, the question is whether we can
partition the vertices of $G$ into $r$ parts such that (a) each part contains
one terminal and (b) there are at most $\ell$ edges with only one endpoint in
this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$
of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time
$2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$
algorithm, where $h$ is the order of the host graph $H$. We also prove that
\textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell
r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called
{\sc List Allocation}, whose expressive power permits the design of
parameterized reductions of both aforementioned problems to it. Then our
results are based on an {\sf FPT}-algorithm for the {\sc List Allocation}
problem that is designed using a suitable adaptation of the {\em randomized
contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk,
and Pilipczuk, FOCS 2012]).
[Show abstract][Hide abstract] ABSTRACT: The tree-cut width of a graph is a graph parameter defined by Wollan [J.
Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut
decompositions. In certain cases, tree-cut width appears to be more adequate
than treewidth as an invariant that, when bounded, can accelerate the
resolution of intractable problems. While designing algorithms for problems
with bounded tree-cut width, it is important to have a parametrically tractable
way to compute the exact value of this parameter or, at least, some constant
approximation of it. In this paper we give a parameterized 2-approximation
algorithm for the computation of tree-cut width; for an input $n$-vertex graph
$G$ and an integer $w$, our algorithm either confirms that the tree-cut width
of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying
that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$.
Prior to this work, no constructive parameterized algorithms, even approximated
ones, existed for computing the tree-cut width of a graph. As a consequence of
the Graph Minors series by Robertson and Seymour, only the existence of a
decision algorithm was known.
[Show abstract][Hide abstract] ABSTRACT: We consider the class of graphs having linear rank-width one, also known as
thread graphs, and investigate a related graph modification problem called the
Thread Vertex Deletion. In this problem, given an $n$ vertex graph $G$ and a
positive integer $k$, we want to decide whether there is a set of at most $k$
vertices whose removal turns $G$ into a thread graph and if one exists, find
such a vertex set. While the meta-theorem of Courcelle, Makowsky, Rotics
implies that Thread Vertex Deletion can be computed in time $f(k)\cdot n^3$, it
is not clear whether this problem allows a runtime with a modest exponential
function. We establish that Thread Vertex Deletion can be solved in time
$8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle
a long induced cycle as an obstruction. To fix this issue, we define a graph
class called the necklace graphs and investigate its structural properties. We
also show that the Thread Vertex Deletion has a polynomial kernel.
[Show abstract][Hide abstract] ABSTRACT: The Hadwiger number of a graph G is the largest integer h such that G has the
complete graph K_h as a minor. We show that the problem of determining the
Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved
in polynomial time on cographs and on bipartite permutation graphs. We also
consider a natural generalization of this problem that asks for the largest
integer h such that G has a minor with h vertices and diameter at most $s$. We
show that this problem can be solved in polynomial time on AT-free graphs when
s>=2, but is NP-hard on chordal graphs for every fixed s>=2.
[Show abstract][Hide abstract] ABSTRACT: The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an
integer $D$, whether it is possible to add edges to $G$ in a way that the
resulting graph is outerplanar and has diameter at most $D$. We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.
[Show abstract][Hide abstract] ABSTRACT: Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for r-Dominating Set and r-Scattered Set on
apex-minor-free graphs, and for Planar-F-Deletion on graphs excluding a fixed
(topological) minor in the case where all the graphs in F are connected.
[Show abstract][Hide abstract] ABSTRACT: Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of level-k networks. In particular, we give an equivalence theorem between circular split systems and unrooted level-1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted level-k phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.
Journal of Bioinformatics and Computational Biology 08/2012; 10(4):1250004. DOI:10.1142/S0219720012500047 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A \emph{$t$-treewidth-modulator} of a graph $G$ is a set $X \subseteq V(G)$
such that the treewidth of $G-X$ is at most some constant $t-1$. In this paper,
we present a novel algorithm to compute a decomposition scheme for graphs $G$
that come equipped with a $t$-treewidth-modulator. This decomposition, called a
\emph{protrusion decomposition}, is the cornerstone in obtaining the following
two main results.
We first show that any parameterized graph problem (with parameter $k$) that
has \emph{finite integer index} and is \emph{treewidth-bounding} admits a
linear kernel on $H$-topological-minor-free graphs, where $H$ is some arbitrary
but fixed graph. A parameterized graph problem is called treewidth-bounding if
all positive instances have a $t$-treewidth-modulator of size $O(k)$, for some
constant $t$. This result partially extends previous meta-theorems on the
existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS
2009] and $H$-minor-free graphs [Fomin et al., SODA 2010].
Our second application concerns the Planar-$\mathcal{F}$-Deletion problem.
Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one
planar graph. Given an $n$-vertex graph $G$ and a non-negative integer $k$,
Planar-$\mathcal{F}$-Deletion asks whether $G$ has a set $X\subseteq V(G)$ such
that $|X|\leq k$ and $G-X$ is $H$-minor-free for every $H\in \mathcal{F}$. Very
recently, an algorithm for Planar-$\mathcal{F}$-Deletion with running time
$2^{O(k)} n \log^2 n$ (such an algorithm is called \emph{single-exponential})
has been presented in [Fomin et al., FOCS 2012] under the condition that every
graph in $\mathcal{F}$ is connected. Using our algorithm to construct
protrusion decompositions as a building block, we get rid of this connectivity
constraint and present an algorithm for the general
Planar-$\mathcal{F}$-Deletion problem running in time $2^{O(k)} n^2$.
[Show abstract][Hide abstract] ABSTRACT: A circle graph is the intersection graph of a set of chords in a circle. Keil
[Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set,
Connected Dominating Set, and Total Dominating Set are NP-complete in circle
graphs. To the best of our knowledge, nothing was known about the parameterized
complexity of these problems in circle graphs. In this paper we prove the
following results, which contribute in this direction:
- Dominating Set, Independent Dominating Set, Connected Dominating Set, Total
Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs,
parameterized by the size of the solution.
- Whereas both Connected Dominating Set and Acyclic Dominating Set are
W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set
is polynomial-time solvable in circle graphs.
- If T is a given tree, deciding whether a circle graph has a dominating set
isomorphic to T is NP-complete when T is in the input, and FPT when
parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.
Theory of Computing Systems 05/2012; 54(1). DOI:10.1007/s00224-013-9478-8 · 0.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given an input graph G and an integer k, the parameterized K_4-minor cover
problem asks whether there is a set S of at most k vertices whose deletion
results in a K_4-minor-free graph, or equivalently in a graph of treewidth at
most 2. This problem is inspired by two well-studied parameterized vertex
deletion problems, Vertex Cover and Feedback Vertex Set, which can also be
expressed as Treewidth-t Vertex Deletion problems: t=0 for Vertex Cover and t=1
for Feedback Vertex Set. While a single-exponential FPT algorithm has been
known for a long time for \textsc{Vertex Cover}, such an algorithm for Feedback
Vertex Set was devised comparatively recently. While it is known to be unlikely
that Treewidth-t Vertex Deletion can be solved in time c^{o(k)}.n^{O(1)}, it
was open whether the K_4-minor cover problem could be solved in
single-exponential FPT time, i.e. in c^k.n^{O(1)} time. This paper answers this
question in the affirmative.
Journal of Computer and System Sciences 04/2012; 81(1). DOI:10.1007/978-3-642-31155-0_11 · 1.14 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several well-studied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict
Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection R\mathcal{R} of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R\mathcal{R}. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertex-kernel. This result improves the best known bound of O(k 2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertex-kernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection R\mathcal{R} of betweenness triplets and seeks an ordering containing all but at most k triplets from R\mathcal{R}. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.
Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings; 11/2011
[Show abstract][Hide abstract] ABSTRACT: Some of the most well studied problems in algorithmic graph theory deal with modifying a graph into an acyclic graph or into a path, using as few operations as possible. In Feedback Vertex Set and Longest Induced Path, the only allowed operation is vertex deletion, and in Spanning Tree and Longest Path, only edge deletions are permitted. We study the edge contraction variant of these problems: given a graph G and an integer k, decide whether G can be transformed into an acyclic graph or into a path using at most k edge contractions. Both problems are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. On the negative side, both problems remain NP-complete when restricted to bipartite input graphs.
[Show abstract][Hide abstract] ABSTRACT: Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely un-explored. We consider two basic edge contraction problems, which we call Path-Contractibility and Tree-Contractibility. Both prob-lems take an undirected graph G and an integer k as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most k edges of G. Our main contribu-tion is an algorithm with running time 4 k+O(log 2 k) + n O(1) for Path-Contractibility and an algorithm with running time 4.88 k n O(1) for Tree-Contractibility, based on a novel application of the color cod-ing technique of Alon, Yuster and Zwick. Furthermore, we show that Path-Contractibility has a kernel with at most 5k + 3 vertices, while Tree-Contractibility does not have a polynomial kernel unless coNP ⊆ NP/poly. We find the latter result surprising, because of the strong connection between Tree-Contractibility and Feedback Vertex Set, which is known to have a vertex kernel with size O(k 2).
[Show abstract][Hide abstract] ABSTRACT: The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.
[Show abstract][Hide abstract] ABSTRACT: Circle graphs are the intersection graphs of chords in a circle. This paper
presents the first sub-quadratic recognition algorithm for the class of circle
graphs. Our algorithm is O(n + m) times the inverse Ackermann function,
{\alpha}(n + m), whose value is smaller than 4 for any practical graph. The
algorithm is based on a new incremental Lexicographic Breadth-First Search
characterization of circle graphs, and a new efficient data-structure for
circle graphs, both developed in the paper. The algorithm is an extension of a
Split Decomposition algorithm with the same running time developed by the
authors in a companion paper.