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Approximation Algorithms for Covering Vertices by Long Paths

Authors Mingyang Gong, Jing Fan, Guohui Lin , Eiji Miyano



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Author Details

Mingyang Gong
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Jing Fan
  • Department of Computing Science, University of Alberta, Edmonton, Canada
  • College of Arts and Sciences, Shanghai Polytechnic University, Shanghai, China
Guohui Lin
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Eiji Miyano
  • Department of Artificial Intelligence, Kyushu Institute of Technology, Iizuka, Japan

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Mingyang Gong, Jing Fan, Guohui Lin, and Eiji Miyano. Approximation Algorithms for Covering Vertices by Long Paths. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 53:1-53:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.53

Abstract

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least k vertices is considered long. When k ≤ 3, the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed k ≥ 4, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when k = 4, the problem admits a 4-approximation algorithm which was presented recently. We propose the first (0.4394 k + O(1))-approximation algorithm for the general problem and an improved 2-approximation algorithm when k = 4. Both algorithms are based on local improvement, and their performance analyses are done via amortization.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • Path cover
  • k-path
  • local improvement
  • amortized analysis
  • approximation algorithm

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