- Chris Schwiegelshohn
- Euiwoong Lee
- Fabrizio Grandoni
- Vincent Pierre Cohen-addad

## Abstract

The $k$-Median problem is one the most fundamental clustering problems: Given $n$ points in a metric space and an integer parameter $k$ our goal is to select precisely $k$ points (called centers) so as to minimize the sum of the distances from each point to the closest center. Obtaining better and better approximation algorithms for $k$-Median is a central open problem.

Over the years, two main approaches have emerged: the first one is the standard Local Search heuristic, yielding a $(3+\eps)$ approximation for any constant $\eps>0$ which is known to be tight. The second approach is based on a Lagrangian Multiplier Preserving (LMP) $\alpha$ approximation algorithm for the related facility location problem. This algorithm is used to build an $\alpha$ approximate fractional bipoint solution for $k$-Median, which is then rounded via a $\rho$ approximate bipoint rounding algorithm. Altogether this gives an $\alpha\cdot \rho$ approximation. A lot of progress was made on improving $\rho$, from the initial $2$ by Jain and Vazirani [FOCS'99, J.ACM'01], to $1.367$ by Li and Svensson [STOC'13, SICOMP'16], and finally to $1.338$ by Byrka et al. [SODA'15, TALG'17]. However for almost 20 years no progress was made on $\alpha$, where the current best result is the classical LMP $2$ approximation algorithm JMS for facility location by Jain et al. [STOC'02, \fab{J.ACM'03}] based on the Dual-Fitting technique.

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