Metric $1$-median selection: Query complexity vs. approximation ratio

TL;DR

This paper establishes lower bounds on the query complexity for approximating the metric 1-median problem within certain ratios, showing that achieving better approximations requires significantly more queries.

Contribution

It proves that no deterministic algorithms can achieve specific approximation ratios with fewer than a certain number of queries, highlighting fundamental limits in metric median approximation.

Findings

No deterministic o(n^{1+1/(h-1)})-query algorithms achieve a (2h - Ω(1))-approximation.

Lower bounds depend on the approximation ratio and query complexity.

Results apply to general metric spaces, indicating inherent computational hardness.

Abstract

Consider the problem of finding a point in a metric space $(\{1,2,\ldots,n\},d)$ with the minimum average distance to other points. We show that this problem has no deterministic $o(n^{1+1/(h-1)})$-query $(2h-\Omega(1))$-approximation algorithms for any constant $h\in\mathbb{Z}^+\setminus\{1\}$.