A Constant-Factor Approximation Algorithm for the k-MST Problem

Given an undirected graph with nonnegative edge costs and an integerk, thek-MST problem is that of finding a tree of minimum cost onknodes. This problem is known to be NP-hard. We present a simple approximation algorithm that finds a solution whose cost is less than 17 times the cost of the optimum. This improves upon previous performance ratios for this problem ?O(k) due to Raviet al.,O(log2k) due to Awerbuchet al., and the previous best bound ofO(logk) due to Rajagopalan and Vazirani. Given any 0<?<1, we first present a bicriteria approximation algorithm that outputs a tree onp??kvertices of total cost at most 2pL/(1??)k, whereLis the cost of the optimalk-MST. The running time of the algorithm isO(n2log2n) on ann-node graph. We then show how to use this algorithm to derive a constant factor approximation algorithm for thek-MST problem. The main subroutine in our algorithm is an approximation algorithm of Goemans and Williamson for the prize-collecting Steiner tree problem.