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 (0.4394k +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.
2012 ACM Subject Classi cation Theory of computation ! Packing and covering problems