R. Ravit Given an undirected graph with non-negative 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(w) due to Ravi et al., 0(log2 k) due to Awerbuch et al, and the previous best bound of O(log k) due to Rajagopalan and Vazirani. Given any O < cr < 1, we first present a bicriteria approximation algorithm that ~o~tputs a tree on p z cYk vertices of total cost at most ~1~, where L is the cost of the optimal kMST. The running time of the algorithm is 0(rz2 log2 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 identical to an approximation algorithm of Goemans and Williamson for the prize-collecting Steiner tree problem. ● School of Computer Science, Carnegie Mellon University, Pittsburgh PA 15213. Supported in part by NSF National Young Investigator grant CCR-9357793 and a Sloan Foundation Research Fellowship. Em ail: avrim@cs.cmu .edu. * Graduate School Of Industrial Administration, Carnegie Mellon University, Pittsburgh PA 15213. Em ail: ravi+@cmu .edu. ~ School of Computer Science, Carnegie Mellon university, Pittsburgh PA 15213. Em ail: svempala@cs.cmu .edu. Permission to make digital/hard copies of all or pari of tlds material for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, tbe c~yright notice, the title of the publication and ita date appear, and notice la given that copyright ia by permission of the ACM, he. To copy otherwise, to republish, to poet on servers or to redistribute to lists, requires specific permission andlor fm. STOC’96, Philadelphia PA, USA @1996 ACM ()-89791.785+/96/()5. .$3.50 Santosh Vempala*
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