A Primal-Dual Approximation Algorithm for the Steiner Forest Problem

Given an undirected graph with nonnegative edge-costs, a subset of nodes of size k called the terminals, and an integer q between 1 and k, the minimum q-Steiner forest problem is to find a forest of minimum cost with at most q trees that spans all the terminals. When q = 1, we have the classical minimum-cost Steiner tree problem on networks. We adapt a primal-dual approximation algorithm for the latter problem due to Agrawal, Klein and Ravi to provide one for the former. The algorithm runs in time 0 (n log n + m ) and outputs a solution of cost at most 2 ( 1 l/ (k q + 1) ) times the value of a lower bound on the cost of any solution. Here n and m denote respectively the number of nodes and edges in the input graph.