A Dependent LP-Rounding Approach for the k-Median Problem

In this paper, we revisit the classical k-median problem: Given n points in a metric space, select k centers so as to minimize the sum of distances of points to their closest center. 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. Our algorithm draws inspiration from clustering and randomized rounding approaches that have been used previously for k-median and the closely related facility location problem, although ensuring that we choose at most k centers requires a careful dependent rounding procedure. 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], our approach is interesting because: (1) The random choice of the k centers given by the algorithm keeps the marginal distributions and satisfies the negative correlation, leading to 3.25 approximation algorithms for some generalizations of the k-median problem, including the k-UFL problem introduced in [8], (2) our algorithm runs in Õ(kn) time compared to the O(n) 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 by a suitable instantiation of various parameters in the algorithm. We also give a 34-approximation for the knapsack median problem, which greatly improves the approximation constant in [11]. Besides the improved approximation ratio, both our algorithm and analysis are simple, compared to [11]. Using the same technique, we also give a 9-approximation for matroid median problem introduced in [9], improving on their 16-approximation.