A Constant-Factor Approximation Algorithm for thek-MST Problem

Abstract

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<α<1, we first present a bicriteria approximation algorithm that outputs a tree onp⩾αkvertices of total cost at most 2pL/(1−α)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.