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

Given an undirected graph with non negative edge costs and an inte ger k the k MST problem is that of nding a tree of minimum cost on k nodes This problem is known to be NP hard We present a simple approximation algorithm that nds a solution whose cost is less than times the cost of the optimum This improves upon previous performance ratios for this problem O p k due to Ravi et al O log k due to Awer buch et al and the previous best bound of O log k due to Rajagopalan and Vazirani Given any we rst present a bicriteria approxi mation algorithm that outputs a tree on p k vertices of total cost at most pL k where L is the cost of the optimal k MST The running time of the algorithm is O n log n on an n node graph We then show how to use this algorithm to derive a constant factor approximation algorithm for the k MST problem The main subroutine in our algorithm is an approx imation algorithm of Goemans and Williamsom for the prize collecting Steiner tree problem School of Computer Science Carnegie Mellon University Pittsburgh PA Supported in part by NSF National Young Investigator grant CCR and a Sloan Foundation Research Fellowship Email avrim cs cmu edu yGraduate School of Industrial Administration Carnegie Mellon University Pittsburgh PA Supported in part by NSF CAREER grant CCR Email ravi cmu edu School of Computer Science Carnegie Mellon University Pittsburgh PA Email svempala cs cmu edu