Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems

Bui-Xuan, Binh-Minh; Telle, Jan Arne; Vatshelle, Martin

doi:10.1016/j.tcs.2013.01.009

Cited by 73 publications

(135 citation statements)

References 10 publications

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“…Golovach, Kratochvíl and Suchý [19] extended these results to the parameterized setting, showing that the existence of a (σ, ρ)-dominating set of size k, and at most k, are W[1]-complete problems when parameterized by k for any pair of finite sets σ and ρ. In contrast, combining our bounds on mim-width and algorithms of Bui-Xuan, Telle, and Vatshelle [8] we obtain the following. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 94%

“…The cuts in a decomposition of constant mim-width are too complex to allow FPT algorithms for interesting NP-hard problems. Instead, what we get is XP algorithms, for the class of LC-VSVP problems [8] -locally checkable vertex subset and vertex partitioning problems -defined in Section 2.4. For classes of bounded mim-width this gives a common explanation for many classical results in the field of algorithms on special graph classes.…”

Section: Introductionmentioning

confidence: 99%

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A Width Parameter Useful for Chordal and Co-comparability Graphs

Kang

Kwon

Strømme

et al. 2017

WALCOM: Algorithms and Computation

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Belmonte and Vatshelle (TCS 2013) used mim-width, a graph width parameter bounded on interval graphs and permutation graphs, to explain existing algorithms for many domination-type problems on those graph classes. We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs K t K t and K t S t are graphs obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that :• interval graphs are (K 3 S 3 )-free chordal graphs; and (K t S t )-free chordal graphs have mim-width at most t − 1,• permutation graphs are (K 3 K 3 )-free co-comparability graphs; and (K t K t )-free cocomparability graphs have mim-width at most t − 1,• chordal graphs and co-comparability graphs have unbounded mim-width in general.We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time n O(t) on (K t S t )-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless P = N P .We generalize these ideas to bigger graph classes. We introduce a new width parameter simwidth, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding K t K t and K t S t as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

A Width Parameter Useful for Chordal and Co-comparability Graphs

Kang

Kwon

Strømme

et al. 2017

WALCOM: Algorithms and Computation

Self Cite

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“…Theorem 3 also implies that several optimization problems can be solved in polynomial time for 2-DORGs (See [13], [14] for details).…”

Section: Discussionmentioning

confidence: 98%

“…It is shown in [13], [14] that the dominating set problem can be solved in polynomial time for the graph that has a decomposition tree of boolean-width O(log n). Since such a decomposition tree of any ORG can be obtained from the orthogonal ray representation [56], the problem can be solved in polynomial time for ORGs provided that orthogonal ray representations are given.…”

Section: Discussionmentioning

confidence: 99%

Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

Takaoka

Tayu

Ueno

2014

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY An orthogonal ray graph is an intersection graph of horizontal and vertical rays (closed half-lines) in the plane. Such a graph is 3-directional if every vertical ray has the same direction, and 2-directional if every vertical ray has the same direction and every horizontal ray has the same direction. We derive some structural properties of orthogonal ray graphs, and based on these properties, we introduce polynomial-time algorithms that solve the dominating set problem, the induced matching problem, and the strong edge coloring problem for these graphs. We show that for 2-directional orthogonal ray graphs, the dominating set problem can be solved in O(n 2 log 5 n) time, the weighted dominating set problem can be solved in O(n 4 log n) time, and the number of dominating sets of a fixed size can be computed in O(n 6 log n) time, where n is the number of vertices in the graph. We also show that for 2-directional orthogonal ray graphs, the weighted induced matching problem and the strong edge coloring problem can be solved in O(n 2 + m log n) time, where m is the number of edges in the graph. Moreover, we show that for 3-directional orthogonal ray graphs, the induced matching problem can be solved in O(m 2 ) time, the weighted induced matching problem can be solved in O(m 4 ) time, and the strong edge coloring problem can be solved in O(m 3 ) time. We finally show that the weighted induced matching problem can be solved in O(m 6 ) time for orthogonal ray graphs.

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“…See [1]- [4] for details. We expect that the complexity of the problems can be reduced by using direct dynamic programming approaches, as shown in this paper.…”

Section: Introductionmentioning

confidence: 99%

Dominating Sets in Two-Directional Orthogonal Ray Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Satoshi TAYU †b) , Members, and Shuichi UENO †c) , Fellow SUMMARY A 2-directional orthogonal ray graph is an intersection graph of rightward rays (half-lines) and downward rays in the plane. We show a dynamic programming algorithm that solves the weighted dominating set problem in O(n 3 ) time for 2-directional orthogonal ray graphs, where n is the number of vertices of a graph.

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Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems

Cited by 73 publications

References 10 publications

A Width Parameter Useful for Chordal and Co-comparability Graphs

A Width Parameter Useful for Chordal and Co-comparability Graphs

Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

Dominating Sets in Two-Directional Orthogonal Ray Graphs

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