Approximation Algorithms for Covering Vertices by Long Paths

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 \(k \le 3\), 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 \(k \ge 4\), 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.