PTAS for minimum k-path vertex cover in ball graph

Zhang, Zhao; Li, Xiaoting; Shi, Yishuo; Nie, Hongmei; Zhu, Yixin

doi:10.1016/j.ipl.2016.11.003

Cited by 14 publications

(5 citation statements)

References 24 publications

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“…It was known that Vertex Cover admits PTASes on fat-object graphs [55,134,144]. Zhang et al [142] showed that the P k -hitting problem (also known as Path Transversal) admits a PTAS on unit-ball graphs. Very recently, Lokshtanov et al [100] gave a general approach which leads to EPTASes for various cases of Subgraph Hitting on disk graphs when the forbidden list F contains a "triangle-bundle" and satisfies the so-called "clone-closed" property.…”

Section: Applications To Broad Graph Classesmentioning

confidence: 99%

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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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Section: Applications To Broad Graph Classesmentioning

confidence: 99%

“…Other approximation algorithms for this problem can be found in [26,141]. We note that the problem admits an EPTAS on disk graphs [100] for 3 ≤ k ≤ 5, and a PTAS on ball graphs for any fixed k [143]. Exact exponential-time algorithms for k = 3 were developed in [86,26,25,139].…”

Section: Applicationsmentioning

confidence: 99%

Reconfiguration on sparse graphs

Lokshtanov

Mouawad

Panolan

et al. 2018

Journal of Computer and System Sciences

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“…A unit disk graph is a disk graph in which all disks have the same radii. For fixed integer k, Zhang et al [80] present a PTAS for MinVCP k on a ball graph whose heterogeneity is bounded by a constant.…”

Section: Theorem 42 [45]mentioning

confidence: 99%

A Survey on the k-Path Vertex Cover Problem

Tu¹

2022

Preprint

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Given a graph G = (V, E) and a positive integer k ≥ 2, a k-path vertex cover is a subset of vertices F such that every path on k vertices in G contains at least one vertex from F . A minimum k-path vertex cover in G is a k-path vertex cover with minimum cardinality and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, it is required to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.

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“…A disk graph is a 2-dimensional ball graph, and if its all disks have the same radii, it is called a unit disk graph. Zhang et al [38] proposed a PTAS for MinVCP k in a ball graph with the ratio r max /r min ≤ c for a constant c.…”

Section: Approximation Algorithmsmentioning

confidence: 99%

A Survey on the k-Path Vertex Cover Problem

2022

Axioms

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Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. A minimum k-path vertex cover in G is a k-path vertex cover having the smallest possible number of vertices and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, the goal is to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.

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PTAS for minimum k-path vertex cover in ball graph

Cited by 14 publications

References 24 publications

Reconfiguration on sparse graphs

Reconfiguration on sparse graphs

A Survey on the k-Path Vertex Cover Problem

A Survey on the k-Path Vertex Cover Problem

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