A Constant-Factor Approximation Algorithm for k-MST Problem

Abstract

Given an undirected graph with nonnegative edge costs and an integer k, the k-MST problem is that of finding a tree of minimum cost on k nodes. 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 Ravi et al., O(log k) due to Awerbuch et al., and the previous best bound of O(log k) due to Rajagopalan and Vazirani. Given any 0<:<1, we first present a bicriteria approximation algorithm that outputs a tree on p :k vertices of total cost at most 2pL (1&:)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 approximation algorithm of Goemans and Williamson for the prize-collecting Steiner tree problem. ] 1999

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@inproceedings{Blum1999ACA, title={A Constant-Factor Approximation Algorithm for k-MST Problem}, author={Avrim Blum and R. Ravi and Santosh Vempala}, year={1999} }