Approximation Algorithms for Covering Vertices by Long Paths

• Published in: Algorithmica 2024-08, Vol.86 (8), p.2625-2651
• Main Authors: Gong, Mingyang, Edgar, Brett, Fan, Jing, Lin, Guohui, Miyano, Eiji
• Format: Article
• Language: English
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Approximation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Mathematics of Computing; Polynomials; Theory of Computation
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-024-01242-3

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seems to escape 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 theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies.