Complexity of the Maximum k-Path Vertex Cover Problem

Eiji Miyano; Toshiki Saitoh; Ryuhei Uehara; Tsuyoshi Yagita; Tom C. van der Zanden

doi:10.1587/transfun.2019DMP0014

Complexity of the Maximum k-Path Vertex Cover Problem

Eiji Miyano^*, Toshiki Saitoh, Ryuhei Uehara, Tsuyoshi Yagita, Tom C. van der Zanden

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

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Abstract

This paper introduces the maximization version of the k-path vertex cover problem, called the Maximum K-Path Vertex Cover problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.

Original language	English
Pages (from-to)	1193-1201
Number of pages	9
Journal	Ieice Transactions on Fundamentals of Electronics Communications and Computer Sciences
Volume	E103A
Issue number	10
DOIs	https://doi.org/10.1587/transfun.2019DMP0014
Publication status	Published - Oct 2020

Keywords

Maximum k-path vertex cover
NP-hardness
polynomial time algorithm
split graphs
bounded treewidth
APPROXIMATION ALGORITHM
FPT ALGORITHM

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10.1587/transfun.2019DMP0014

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Cite this

@article{a6298ba43606428a9334f8d45eb37304,

title = "Complexity of the Maximum k-Path Vertex Cover Problem",

abstract = "This paper introduces the maximization version of the k-path vertex cover problem, called the Maximum K-Path Vertex Cover problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.",

keywords = "Maximum k-path vertex cover, NP-hardness, polynomial time algorithm, split graphs, bounded treewidth, APPROXIMATION ALGORITHM, FPT ALGORITHM",

author = "Eiji Miyano and Toshiki Saitoh and Ryuhei Uehara and Tsuyoshi Yagita and {van der Zanden}, {Tom C.}",

note = "data source: no data used",

year = "2020",

month = oct,

doi = "10.1587/transfun.2019DMP0014",

language = "English",

volume = "E103A",

pages = "1193--1201",

journal = "Ieice Transactions on Fundamentals of Electronics Communications and Computer Sciences",

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TY - JOUR

T1 - Complexity of the Maximum k-Path Vertex Cover Problem

AU - Miyano, Eiji

AU - Saitoh, Toshiki

AU - Uehara, Ryuhei

AU - Yagita, Tsuyoshi

AU - van der Zanden, Tom C.

N1 - data source: no data used

PY - 2020/10

Y1 - 2020/10

N2 - This paper introduces the maximization version of the k-path vertex cover problem, called the Maximum K-Path Vertex Cover problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.

AB - This paper introduces the maximization version of the k-path vertex cover problem, called the Maximum K-Path Vertex Cover problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.

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KW - FPT ALGORITHM

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