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

arXiv: https://arxiv.org/abs/1711.08715 · HTML: https://mlsanity.com/paper/1711.08715/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2017.

## Built on by (most cited first, 6 of 6)

- [Better Algorithms for Individually Fair k-Clustering](https://mlsanity.com/paper/2106.12150/) (2021)
- [A Scalable Algorithm for Individually Fair K-means Clustering](https://mlsanity.com/paper/2402.06730/) (2024)
- [Optimizing Multiple Simultaneous Objectives for Voting and Facility Location](https://mlsanity.com/paper/2212.03467/) (2022)
- [Clustering to Minimize Cluster-Aware Norm Objectives](https://mlsanity.com/paper/2410.24104/) (2024)
- [Compatibility of Max and Sum Objectives for Committee Selection and k-Facility Location](https://mlsanity.com/paper/2507.17063/) (2025)
- [A Broader View on Clustering under Cluster-Aware Norm Objectives](https://mlsanity.com/paper/2512.06211/) (2025)

## Abstract (arXiv)

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.