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We give an improved approximation algorithm for the general k-medians problem. Given any \epsilon>0, the algorithm finds a solution of total distance at most D(1+\epsilon) using at most k ln(n+n/\epsilon) medians (a.k.a. sites), provided some solution of total distance D using k medians exists. This improves over the best previous bound (w.r.t. the number of medians) by a factor of \Omega(1/\epsilon) provided 1/\epsilon=n^O(1). The algorithm is a greedy algorithm, derived using the method of oblivious randomized rounding. It requires at most k ln(n+n/\epsilon) linear-time iterations. We also derive algorithms for fractional and weighted variants of the problem.
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Neal E. Young, "Greedy Approximation Algorithms for K-Medians by Randomized Rounding." Dartmouth Computer Science Technical Report PCS-TR99-344, March 1999.
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