Computing a Clique Tree with the Algorithm Maximal Label Search

Berry, Alain; Simonet, Geneviève

doi:10.3390/a10010020

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…First we compute a minimal triangulation G ′ of G; this can be done in time O(n µ log n) [56]. Since G ′ is a chordal graph, we use an algorithm due to Berry and Simonet [11]…”

Section: K 2q -Induced-minor-free Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Dallard

Milanič

Štorgel

2020

Graph-Theoretic Concepts in Computer Science

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We continue the study of (tw, ω)-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Maximum Independent Set and related problems.In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a minmax graph invariant related to tree decompositions. Bounded tree-independence number implies both (tw, ω)-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.In this paper, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a K 5 minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph K 2,q implies the existence of a tree decomposition with independence number at most 2(q − 1).These results are obtained using a variety of tools, including ℓ-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.

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“…First we compute a minimal triangulation G ′ of G; this can be done in time O(n µ log n) [56]. Since G ′ is a chordal graph, we use an algorithm due to Berry and Simonet [11]…”

Section: K 2q -Induced-minor-free Graphsmentioning

confidence: 99%

“…. , v n ) of G. Berry and Simonet gave in [11] a linear-time algorithm that takes as input a connected chordal graph G and a perfect moplex ordering of G, and computes a clique tree T of G. We explain the idea of their algorithm in terms of the perfect moplex partition (M 1 , . .…”

Section: Lemma 52 Let G Be a Chordal Graph With N Vertices And M Edge...mentioning

confidence: 99%

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Dallard

Milanič

Štorgel

2020

Graph-Theoretic Concepts in Computer Science

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“…Recall that a chordal graph always admits a clique-tree decomposition, i.e., a tree decomposition where each bag forms a maximal clique; moreover, such a decomposition can be found in linear time (Berry and Simonet 2017). Hence the treewidth of a chordal graph is bounded by the maximum degree of the graph, and we have the following corollary.…”

Section: Constructing Dependency Graphsmentioning

confidence: 86%

Cooperative Games With Bounded Dependency Degree

Igarashi

Izsak

Elkind

2018

AAAI

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Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of coopera- tive games. We show that many computational problems for cooperative games become tractable for games whose dependency degree or supermodular degree are bounded. In particular, we prove that simple games admit efficient algorithms for various solution concepts when the supermodular degree is small; further, we show that computing the Shapley value is always in FPT with respect to the dependency degree. Finally, we observe that, while determining the dependency among players is computationally hard, there are efficient algorithms for special classes of games.

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“…. , v n ) of G. Berry and Simonet gave in [12] a linear-time algorithm that takes as input a connected chordal graph G and a perfect moplex ordering of G, and computes a clique tree T of G. We explain the idea of their algorithm in terms of the perfect moplex partition (M 1 , . .…”

mentioning

confidence: 99%

Treewidth versus clique number. II. Tree-independence number

Dallard¹,

Milanič²,

Štorgel³

2021

Preprint

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In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw, ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether (tw, ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent H-Packing problem, for any fixed finite set H of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw, ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of...

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Computing a Clique Tree with the Algorithm Maximal Label Search

Cited by 5 publications

References 26 publications

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Cooperative Games With Bounded Dependency Degree

Treewidth versus clique number. II. Tree-independence number

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