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 $(mathcal{D},{c_{ij}})$ with $n=|mathcal{D}|$ points, and a non-increasing weight vector $winmathbb{R}_+^n$, and the goal is to open $k$ centers and assign each point each point $jinmathcal{D}$ to a center so as to minimize $w_1cdottext{(largest assignment cost)}+w_2cdottext{(second-largest assignment cost)}+ldots+w_ncdottext{($n$-th largest assignment cost)}$. We give an $(18+epsilon)$-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 $ell$ 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 $(8.5+epsilon)$-approximation algorithm for ${0,1}$ weights.