Approximate association via dissociation

Jie, You; Wang, Jianxin; Cao, Yixin

doi:10.1016/j.dam.2016.11.007

Cited by 14 publications

(7 citation statements)

References 22 publications

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“…In the special case of the Max Weight Independent Packing when H = H(G, F), we obtain the Max Weight Independent F-Packing problem, a common generalization of several problems studied in the literature, including the Independent F-Packing problem (see [22]), the Max Weight Independent Set problem, the Max Weight Induced Matching problem (see, e.g., [2,68]), the Dissociation Set problem (see, e.g., [67,82,83]), and the k-Separator problem (see, e.g., [8,61]).…”

Section: Lemma 23 (Corollary 36 Inmentioning

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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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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Section: Lemma 23 (Corollary 36 Inmentioning

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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“…As such, it trivially has a 3-approximation (polynomial-time) algorithm. The first non-trivial approximation algorithm was a 5 3 -approximation due to You et al [140]. Shortly afterward, Fiorini et al gave a 7 3approximation [57], and subsequently a 9 4 -approximation [58].…”

Section: Applicationsmentioning

confidence: 99%

“…In particular, by setting the forbidden set F to be various fixed (finite) sets, it is seen that (Induced) Subgraph Hitting generalizes many fundamental vertex-deletion problems, which were previously extensively studied on their own in the literature, particularly (but not only) from the perspectives of approximation and parameterization. To name a few, these include Vertex Cover, (Induced) P k -Hitting where P k is the path of length k [14,76,96] (which subsumes Cluster Vertex Deletion [2,58,140]), Triangle Hitting [99,100] or more generally (Induced) C k -Hitting where C k is the cycle of length k [74,119], K k -Hitting where K k is the clique of size k [59], (Induced) Biclique Hitting [72], Component Order Connectivity [44,75,91], Degree Modulator [9,71,76], and Treedepth Modulator [12,53,69]. We elaborate on related works, particularly on these specific problems, in Appendix A.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguration on sparse graphs

Lokshtanov

Mouawad

Panolan

et al. 2018

Journal of Computer and System Sciences

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A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k.We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs.

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“…(We defer their definitions to the next section.) Many algorithms have been developed for vertex deletion problems to chordal graphs and its subclasses,-most notably (unit) interval graphs, cluster graphs, and split graphs; see, e.g., [17,4,10,9,8,34,12,25, 1] for a partial list. After the long progress of algorithmic achievements, some natural questions arise: What is the complexity of transforming a chordal graph to a (unit) interval graph, a cluster graph, a split graph, or a member of some other subclass of chordal graphs?…”

mentioning

confidence: 99%

“…However, our main motivation is from the recent algorithmic progress in vertex deletion problems. It has come to our attention that to transform a graph to class C 1 , it is frequently convenient to first make it a member of another class C 2 that contains C 1 as a proper subclass, followed by an algorithm for the C 2 → C 1 problem [30,9,7,34].…”

mentioning

confidence: 99%

Vertex deletion problems on chordal graphs

Cao

Otachi

et al. 2018

Theoretical Computer Science

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Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yannakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. IntroductionGenerally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. Many classic optimization problems belong to the family of vertex deletion problems, and their algorithms and complexity have been intensively studied. For example, the clique problem and the independent set problem are nothing but the vertex deletion problems to complete graphs and to edgeless graphs respectively. Most interesting graph properties are hereditary: If a graph satisfies this property, then so does every induced subgraph of it. For all the vertex deletion problems to hereditary graph classes, Lewis and Yannakakis [27] have settled their complexity once and for all with a dichotomy result: They are either NP-hard or trivial. Thereafter algorithmic efforts were mostly focused on the nontrivial ones, and the major approaches include approximation algorithms [28], parameterized algorithms [6], and exact algorithms [15].Chordal graphs make one of the most important graph classes. Together with many of its subclasses, it has played important roles in the development of structural graph theory. (We defer their definitions to the next section.) Many algorithms have been developed for vertex deletion problems to chordal graphs and its subclasses,-most notably (unit) interval graphs, cluster graphs, and split graphs; see, e.g., [17,4,10,9,8,34,12,25, 1] for a partial list. After the long progress of algorithmic achievements, some natural questions arise: What is the complexity of transforming a chordal graph to a (unit) interval graph, a cluster graph, a split graph, or a member of some other subclass of chordal graphs? It is quite surprising that this type of problems has not been systematically studied, save few concrete results, e.g., the polynomial-time algorithms for the clique problem, the independent set problem, and the feedback vertex set problem (the object class being forests) [21,33].The same question can be asked for other pair of source and object graph classes. The most important source classes include planar graphs [20,18,16], bipartite graphs [32], and degree-bounded graphs [19]. As one may expect, with special properties imposed on input graphs, the problems become easier, and some of them may not remain NP-hard. Unfortunately, a clear-cut answer to them seems very unlikely, since their complexity would...

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Approximate association via dissociation

Cited by 14 publications

References 22 publications

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Reconfiguration on sparse graphs

Vertex deletion problems on chordal graphs

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