Complexity of Most Vital Nodes for Independent Set in Graphs Related to Tree Structures

Bazgan, Cristina; Toubaline, Sonia; Tuza, Źsolt

doi:10.1007/978-3-642-19222-7_17

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…which proves that k Most Vital Nodes 1-median is NP -hard to approximate within a factor 3 2 − ǫ, for any ǫ > 0. First observe that there exists at least one optimal solution of k Most Vital Nodes 1-median containing only nodes of V .…”

Section: Np-hardness Of Approximationmentioning

confidence: 81%

“…∈ E which is not covered. This implies in G ′ the existence of a path from x i (or x ′ i ) to x j , with i < j, passing through the gadget corresponding to the edge ( 3 and |V | = n. We construct an instance I ′ of k Most Vital Nodes 1-median consisting of a graph G ′ = (V ′ , E ′ ) with k < n as follows (see Figure 3). G ′ is a copy of G to which we add complete graphs K i n with n nodes x 1 i , .…”

Section: Np-hardness Of Approximationmentioning

confidence: 99%

“…For maximum matching, min edge blocker is NP -hard even for bipartite graphs [12], but polynomial for grids and trees [10]. In [3], the k most vital nodes and min node blocker versions with respect to independent set for bipartite graphs remain polynomial on the unweighted graphs and become NP -hard for weighted graphs. For bounded treewidth graphs and cographs these versions remain polynomial [3].…”

Section: Introductionmentioning

confidence: 99%

“…In [3], the k most vital nodes and min node blocker versions with respect to independent set for bipartite graphs remain polynomial on the unweighted graphs and become NP -hard for weighted graphs. For bounded treewidth graphs and cographs these versions remain polynomial [3]. Concerning the approximation on bipartite weighted graphs, k most vital nodes with respect to independent set has no ptas [3].…”

Section: Introductionmentioning

confidence: 99%

“…For bounded treewidth graphs and cographs these versions remain polynomial [3]. Concerning the approximation on bipartite weighted graphs, k most vital nodes with respect to independent set has no ptas [3].…”

Section: Introductionmentioning

confidence: 99%

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Complexity of determining the most vital elements for the p-median and p-center location problems

2012

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We consider the k most vital edges (nodes) and min edge (node) blocker versions of the 1-median and 1-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) 1-median (respectively 1-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the 1-median (respectively 1-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker 1-median (respectively 1-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the 1-median (respectively 1-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges 1median and k most vital edges 1-center are NP-hard to approximate within a factor 7 5 − ǫ and 4 3 − ǫ respectively, for any ǫ > 0, while k most vital nodes 1-median and k most vital nodes 1-center are NP-hard to approximate within a factor 3 2 − ǫ, for any ǫ > 0. We also show that the complementary versions of these four problems are NP-hard to approximate within a factor 1.36.

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Section: Np-hardness Of Approximationmentioning

confidence: 81%

Section: Np-hardness Of Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complexity of determining the most vital elements for the p-median and p-center location problems

2012

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Contraction Blockers for Graphs with Forbidden Induced Paths

Diner

Paulusma

Picouleau

et al. 2015

Lecture Notes in Computer Science

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hinerD ¤ yF nd ulusm D hF nd i oule uD gF nd iesD fF @PHISA 9gontr tion lo kers for gr phs with for idden indu ed p thsF9D in elgorithms nd omplexity X Wth sntern tion l gonferen eD gseg PHISD risD pr n eD w y PHEPPD PHIS Y pro eedingsF D ppF IWREPHUF ve ture notes in omputer s ien eF @WHUWAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-18173-814.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pfree graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomialtime solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.

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Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Paulusma

Picouleau

Ries

2016

Lecture Notes in Computer Science

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H.

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Complexity of Most Vital Nodes for Independent Set in Graphs Related to Tree Structures

Cited by 4 publications

References 13 publications

Complexity of determining the most vital elements for the p-median and p-center location problems

Complexity of determining the most vital elements for the p-median and p-center location problems

Contraction Blockers for Graphs with Forbidden Induced Paths

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

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