Abstract
In this paper, we revisit the classical k-median problem. Using the standard LP relaxation for k-median, we give an efficient algorithm to construct a probability distribution on sets of k centers that matches the marginals specified by the optimal LP solution. Analyzing the approximation ratio of our algorithm presents significant technical difficulties: we are able to show an upper bound of 3.25. While this is worse than the current best known 3 + ε guarantee of [2], because: (1) it leads to 3.25 approximation algorithms for some generalizations of the k-median problem, including the k-facility location problem introduced in [10], (2) our algorithm runs in \(\tilde{O}(k^3 n^2/\delta^2)\) time to achieve 3.25(1 + δ)-approximation compared to the O(n 8) time required by the local search algorithm of [2] to guarantee a 3.25 approximation, and (3) our approach has the potential to beat the decade old bound of 3 + ε for k-median.
We also give a 34-approximation for the knapsack median problem, which greatly improves the approximation constant in [13]. Using the same technique, we also give a 9-approximation for matroid median problem introduced in [11], improving on their 16-approximation.
A full version of this paper is available at the authors’ web pages.
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Charikar, M., Li, S. (2012). A Dependent LP-Rounding Approach for the k-Median Problem. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_17
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DOI: https://doi.org/10.1007/978-3-642-31594-7_17
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