Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Philip, Geevarghese; Raman, Venkatesh; Sikdar, Somnath

doi:10.1145/2390176.2390187

Cited by 73 publications

(42 citation statements)

References 32 publications

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“…Our algorithm to find a k-domination core heavily relies on several earlier results. We combine several of the ideas introduced in [2,10,14,40,41,46] and, as a byproduct, are able to simplify the proofs of some of the results. Note that V (G) is a k-domination core.…”

Section: A General Frameworkmentioning

confidence: 99%

“…For this reason, the (Connected) Dominating Set problem and (Connected) Distance-r Dominating Set problem have been extensively studied on restricted graph classes. A particularly fruitful line of research in this area concerns kernelization algorithms for the aforementioned problems [1,4,17,18,19,41].…”

mentioning

confidence: 99%

“…Philip et al [41] obtained a kernel of size O(k (d+1) 2 ) on d-degenerate graphs, for constant d, and more generally a kernel of size O(k max(i 2 ,j 2 ) ) on graphs excluding the biclique K i,j as a subgraph. On the lower bounds side, Cygan et al [8] have shown that the existence of a size O(k (d−1)(d−3)− ) kernel, > 0, for Dominating Set on d-degenerate graphs would imply NP ⊆ coNP/poly.…”

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confidence: 99%

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Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben¹,

Kumar²,

Mouawad³

et al. 2019

SIAM J. Discrete Math.

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For α > 1, an α-approximate (bi-)kernel is a polynomial-time algorithm that takes as input an instance (I, k) of a problem Q and outputs an instance (I , k ) (of a problem Q ) of size bounded by a function of k such that, for every c ≥ 1, a c-approximate solution for the new instance can be turned into a (c · α)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every α > 1, Connected Dominating Set admits a polynomial-size α-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP ⊆ coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r ∈ N there cannot exist an α-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any α > 1 (assuming the Gap Exponential Time Hypothesis).

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Section: A General Frameworkmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben¹,

Kumar²,

Mouawad³

et al. 2019

SIAM J. Discrete Math.

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“…A polynomial kernel for p-F-Deletion for class F consisting of a graph with two vertices and several parallel edges is given in [47]. Philip et al [68] and Cygan et al [31] obtained polynomial kernels for Pathwidth 1-Deletion. Our next theorem generalizes all these kernelization results.…”

mentioning

confidence: 99%

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Fomin

Lokshtanov²,

Misra

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

130

174

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Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth η-Deletion.In this paper we obtain a number of generic algorithmic results about F-Deletion, when F contains at least one planar graph. The highlights of our work are• A constant factor approximation algorithm for the optimization version of F-Deletion;• A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of F-Deletion where all graphs in F are connected;• A polynomial kernel for parameterized F-Deletion.These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -constant factor approximation, polynomial kernelization and FPT algorithms -are stringed together by a common theme of polynomial time preprocessing.

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“…Thus, we wonder whether there exist deterministic algorithms forPseudoforest Deletion that run in O(c k n) time for constant c depending on . Philip et al [13] show that for every , -Pseudoforest Deletion has a kernel with O(k 2 ) vertices. Given the local nature of Pseudoforest Deletion, it is interesting to see if there exists a kernel for it with a linear number of vertices.…”

Section: Discussionmentioning

confidence: 99%

A faster parameterized algorithm for Pseudoforest Deletion

Bodlaender

Ono

Otachi

2018

Discrete Applied Mathematics

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A pseudoforest is a graph where each connected component contains at most one cycle, or alternatively, a graph that can be turned into a forest by removing at most one edge from each connected component. In this paper, we show that the following problem can be solved in O(3 k nk O(1) ) time: given a graph G and an integer k, can we delete at most k vertices from G such that we obtain a pseudoforest? The result improves upon an earlier result by Philip et al. [MFCS 2015] who gave a (nonlinear) 7.56 k n O(1) -time algorithm both in the exponential factor depending on k as well as in the polynomial factor depending on n. ACM Subject Classification G.2.2 Graph Theory

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Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Cited by 73 publications

References 32 publications

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

A faster parameterized algorithm for Pseudoforest Deletion

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