How to trim a MST

Abstract

The minimum cost-tree cover problem is to compute a minimum cost-tree T in a given connected graph G with costs on the edges, such that the vertices spanned by T form a vertex cover for G . The problem is supposed to occur in applications of vertex cover and in edge-dominating sets when additional connectivity is required for solutions. Whereas a linear-time 2 -approximation algorithm for the unweighted case has been known for quite a while, the best approximation ratio known for the weighted case is 3 . Moreover, the 3 -approximation algorithms for such cases are far from practical due to their inefficiency. In this article we present a fast, purely combinatorial 2 -approximation algorithm for the minimum cost-tree cover problem. It constructs a good approximate solution by trimming some leaves within a minimum spanning tree (MST); and, to determine which leaves to trim, it uses both the primal-dual schema and an instance layering technique adapted from the local ratio method.