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 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 ∈ Rn+, 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) + . . . + wn · (n-th largest assignment cost). We give an (18 + ǫ)-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 + ǫ)-approximation algorithm for {0, 1} weights.