Bounds on the quality of approximate solutions to the group Steiner problem

Abstract

The Group Steiner Problem (GSP) is a generalized version of the well known Steiner Problem. For an undirected, connected distance graph with groups of required vertices and Steiner vertices, GSP asks for a shortest connected subgraph, containing at least one vertex of each group. As the Steiner Problem is NP-hard, GSP is too, and we are interested in approximation algorithms. Efficient approximation algorithms have already been proposed, but nothing about the quality of any approximate solution is known so far. Especially for the VLSI design application of the problem, bounds on the quality of approximate solutions are of great importance. We present a simple polynomial time approximation algorithm that computes a tree with no more than $g-1$ times the length of a minimal tree, where $g$ is the number of required groups. In addition, we propose an extended version of this algorithm, trading quality of the solution for computation time. Here, one extreme is just our proposed approximation, and the other is an optimal solution. Moreover, we will prove the quality bound $g-1$ for a modification of an efficient approximation algorithm proposed in the literature.