Improved Parameterized Upper Bounds for Vertex Cover

Chen, Jianer; Kanj, Iyad A.; Xia, Ge

doi:10.1007/11821069_21

Cited by 107 publications

(93 citation statements)

References 18 publications

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“…Assume that we are given a vertex cover C ⊆ V of size c := vc(G). If not, we can compute it in time O(1.2738 c + cn) by [1]. Each class that contains at least 2 vertices must contain some vertex of C; otherwise, it would not be connected.…”

Section: Now We Present Two Fpt Results For Ecpmentioning

confidence: 99%

“…We show that ECP is W [1]-hard when parameterized by pw(G), f vs(G), and the number of partitions r combined. We show that this result holds true even for planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…These results studied the parameterized complexity of some NP-hard labeling and coloring problems with respect to graph-structural parameters, such as the treewidth, the vertex cover number, and the feedback vertex set number. For example, it was shown in [4] that EC is W [1]-hard when parameterized by tw(G) and r combined. More recently, it was shown in [6] that EC is FPT when parameterized by vc(G).…”

Section: Introductionmentioning

confidence: 99%

“…some parameter such as the solution size), are practically important. The parameterized complexity working hypothesis (FPT = W [1]) implies that these problems are fixed-parameter intractable. Therefore, it is both interesting and important to study whether these problems become parameterized tractable when parameterized by other (tractable) structural parameters.…”

Section: Introductionmentioning

confidence: 99%

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What Makes Equitable Connected Partition Easy

Enciso

Fellows

Guo

et al. 2009

Parameterized and Exact Computation

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Abstract. We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of partitions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minimum feedback vertex set, the size of a minimum vertex cover, and the maximum number of leaves in a spanning tree of the graph. In particular, we show that the problem is W[1]-hard with respect to the first four parameters (even combined), whereas it becomes fixed-parameter tractable when parameterized by the last two parameters. The hardness result remains true even for planar graphs. We also show that the problem is in XP when parameterized by the treewidth (and hence any other mentioned structural parameter). Furthermore, we show that the closely related problem, Equitable Coloring, is FPT when parameterized by the maximum number of leaves in a spanning tree of the graph.

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Section: Now We Present Two Fpt Results For Ecpmentioning

confidence: 99%

“…We show that ECP is W [1]-hard when parameterized by pw(G), f vs(G), and the number of partitions r combined. We show that this result holds true even for planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

What Makes Equitable Connected Partition Easy

Enciso

Fellows

Guo

et al. 2009

Parameterized and Exact Computation

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show abstract

“…Note that currently, for the parameter vertex cover number, the best known fpt-algorithm for the vertex cover problem, by Chen Kanj and Xia [5], runs in time O(1.2738 k + kn) ⊆ O(1.2738 k · n 2 ). This gives an overall running time of O((2k) 2k · 1.2738 k · n 2 ) for our algorithm.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Isomorphism of (mis)Labeled Graphs

Schweitzer

2011

Algorithms – ESA 2011

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Abstract. For similarity measures of labeled and unlabeled graphs, we study the complexity of the graph isomorphism problem for pairs of input graphs which are close with respect to the measure. More precisely, we show that for every fixed integer k we can decide in quadratic time whether a labeled graph G can be obtained from another labeled graph H by relabeling at most k vertices. We extend the algorithm solving this problem to an algorithm determining the number ℓ of vertices that must be deleted and the number k of vertices that must be relabeled in order to make the graphs equivalent. The algorithm is fixed-parameter tractable in k + ℓ. Contrasting these tractability results, we also show that for those similarity measures that change only by finite amount d whenever one edge is relocated, the problem of deciding isomorphism of input pairs of bounded distance d is equivalent to solving graph isomorphism in general.

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