Approximation algorithms for minimum weight connected 3-path vertex cover

摘要

A k-path vertex cover (VCPk) is a vertex set C of graph G such that every path of G on k vertices has at least one vertex in C. Because of its background in keeping data integrality of a network, minimum VCPk problem (MinVCPk) has attracted a lot of researches in recent years. This paper studies the minimum weight connected VCPk problem (MinWCVCPk), in which every vertex has a weight and the VCPk found by the algorithm induces a connected subgraph and has the minimum weight. It is known that MinWCVCPk is set-cover-hard. We present two polynomial-time approximation algorithms for MinWCVCP3. The first one is a greedy algorithm achieving approximation ratio 3ln n. The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a VCP3 in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most lnδmax+4+ln2, where δmax is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight.