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

TL;DR

This paper introduces approximation algorithms for the ordered $k$-median problem, generalizing $k$-median and $k$-center, with a focus on weighted assignment costs and special cases with $ ext{0/1}$ weights.

Contribution

The authors develop an $(18+ ext{epsilon})$-approximation algorithm for ordered $k$-median and a novel reduction for $ ext{0/1}$ weights, linking it to standard $k$-median guarantees.

Findings

Achieved an $(18+ ext{epsilon})$-approximation for ordered $k$-median.

Established a reduction for $ ext{0/1}$ weights to standard $k$-median.

Provided an $(8.5+ ext{epsilon})$-approximation for $ ext{0/1}$ weighted case.

Abstract

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 $w\in\mathbb{R}_+^n$, and the goal is to open $k$ centers and assign each point each point $j\in\mathcal{D}$ to a center so as to minimize $w_1\cdot\text{(largest assignment cost)}+w_2\cdot\text{(second-largest assignment cost)}+\ldots+w_n\cdot\text{($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.