Computing a tree having a small vertex cover

Takuro Fukunaga, Takanori Maehara

Abstract: We consider a new Steiner tree problem, called vertex-cover-weighted Steiner tree problem. This problem defines the weight of a Steiner tree as the minimum weight of vertex covers in the tree, and seeks a minimum-weight Steiner tree in a given vertex-weighted undirected graph. The problem admits an O ( log ⁡ n ) -approximation algorithm in general graphs with n vertices, and this approximation factor is tight up to a constant because it is NP-hard to achieve an o ( log ⁡ n ) -approximation for the vertex-cover-weighted Steiner tree problem in general graphs even if the given vertex weights are uniform and a spanning tree is required. In this paper, we present constant-factor approximation algorithms for the problem in unit disk graphs and in graphs excluding a fixed minor. For the latter graph class, our algorithm can be also applied for the Steiner tree activation problem.