Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median

We consider a generalization of k-median and k-center, called the em ordered k-median problem. In this problem, we are given a metric space (D,cij) with n=|D| points, and a non-increasing weight vector w∈ℝ_+n, and the goal is to open k centers and assign each point each point j∈D to a center so as to minimize w1·(largest assignment cost)+w2·(second-largest assignment cost)+ldots+wn·(n-th largest assignment cost). We give an (18+ε)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of 0,1-weights, which models the problem of minimizing the ℓ largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+ε)-approximation algorithm for 0,1 weights.