The matroid median problem

In the classical k-median problem, we are given a metric space and would like to open k centers so as to minimize the sum (over all the vertices) of the distance of each vertex to its nearest open center. In this paper, we consider the following generalization of the problem: instead of opening at most k centers, what if each center belongs to one of T different types, and we are allowed to open at most ki centers of type i (for each i=1,2,...,T). The case T = 1 is the classical k-median, and the case of T = 2 is the red-blue median problem for which Hajiaghayi et al. [ESA 2010] recently gave a constant-factor approximation algorithm. Even more generally, what if the set of open centers had to form an independent set from a matroid? In this paper, we give a constant factor approximation algorithm for such matroid median problems. Our algorithm is based on rounding a natural LP relaxation in two stages: in the first step, we sparsify the structure of the fractional solution while increasing the objective function value by only a constant factor. This enables us to write another LP in the second phase, for which the sparsified LP solution is feasible. We then show that this second phase LP is in fact integral; the integrality proof is based on a connection to matroid intersection. We also consider the penalty version (alternately, the so-called prize collecting version) of the matroid median problem and obtain a constant factor approximation algorithm for it. Finally, we look at the Knapsack Median problem (in which the facilities have costs and the set of open facilities need to fit into a Knapsack) and get a bicriteria approximation algorithm which violates the Knapsack bound by a small additive amount.