A lower bound for metric 1-median selection

TL;DR

This paper establishes a fundamental lower bound on the query complexity for approximating the metric 1-median problem, showing no efficient deterministic algorithms can achieve better than a 4-approximation with fewer than quadratic queries.

Contribution

It proves a tight lower bound on the query complexity for deterministic approximation algorithms for the metric 1-median problem.

Findings

No deterministic o(n^2)-query algorithms can achieve better than 4-approximation.

The problem requires quadratic query complexity for near-optimal solutions.

Establishes fundamental limits for metric median approximation algorithms.

Abstract

Consider the problem of finding a point in an n-point metric space with the minimum average distance to all points. We show that this problem has no deterministic $o(n^2)$-query $(4-\Omega(1))$-approximation algorithms.