Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

Escoffier, Bruno; Gourvès, Laurent; Monnot, Jérôme

doi:10.1016/j.jda.2009.01.005

Cited by 49 publications

(44 citation statements)

References 26 publications

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“…It is easy to observe that Vertex Cover reduces to Connected Vertex Cover implying that the latter is at least as hard as the former in general graphs. In fact, Connected Vertex Cover is NP-hard even on bipartite graphs [13] where Vertex Cover is solvable in polynomial time (Theorem 2.1.1 [11]). However, Connected Vertex Cover is polynomialtime solvable on chordal graphs and sub-cubic graphs [13,26] and has an O * (2 O(t) ) algorithm when the input graph has treewidth upper bounded by t [2].…”

Section: Introduction Motivation and Our Resultsmentioning

confidence: 99%

Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

2018

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The Connected Vertex Cover problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al. [STOC 2017], have shown an α-approximate kernel for the problem for every α > 1, in the framework of approximate or lossy kernelization. We exhibit lossy kernels and FPT algorithms for Connected Vertex Cover for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size.Our first result is a lossy kernel for Connected Vertex Cover parameterized by the size k of a split deletion set. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set and a split deletion set is a set of vertices whose deletion results in a split graph. Let n denote the number of vertices in the input graph. We show that

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Section: Introduction Motivation and Our Resultsmentioning

confidence: 99%

Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

2018

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“…So, each of the biconnected components of G 1 also has one vertex whose deletion results in a chordal graph. Hence, all the biconnected components of G 1 are also in the graph class H. It is known due to Lemma 4 of Escoffier et al [15] that finding a minimum a connected vertex cover and finding smallest connected vertex cover containing u are polynomially equivalent in G 1 . Now, we explain how Connected Vertex Cover is polynomial-time solvable on G 1 .…”

Section: Cvc-split-cograph-deletionmentioning

confidence: 96%

“…We define H by the class of all connected graphs that contain a vertex whose removal results in a chordal graph. Note that G 1 ∈ H. Suppose that a new vertex w, and an edge wv such that v ∈ V (G 1 ) are added to G 1 (called pendant addition by Escoffier et al [15]). Even then also G 1 − {u} is a chordal graph.…”

Section: Cvc-split-cograph-deletionmentioning

confidence: 99%

On the Approximate Compressibility of Connected Vertex Cover

2020

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The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have a polynomial kernelization algorithm. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokshtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0.In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.

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“…We call this grid drawing D φ . We use the following simple trick from [16] to make an equivalent instance that is a grid graph. Observation 30.…”

Section: Dominating Setmentioning

confidence: 99%

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

Berg

Bodlaender

Kisfaludi-Bak

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2 O(n 1−1/d ) for any fixed dimension d ≥ 2 for many well known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques.The lower bound framework is based on a constructive embedding of graphs into ddimensional grids, and it allows us to derive matching 2 Ω(n 1−1/d ) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

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Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

Cited by 49 publications

References 26 publications

Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

On the Approximate Compressibility of Connected Vertex Cover

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

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