Deterministic metric $1$-median selection with very few queries

TL;DR

This paper introduces a deterministic, sublinear-query algorithm for the metric 1-median problem that achieves near-optimal approximation, and proves limitations of existing algorithms in query complexity and approximation ratio.

Contribution

It presents the first deterministic sublinear-query, sub-logarithmic-approximation algorithm for metric 1-median and establishes lower bounds on the approximation capabilities of deterministic algorithms.

Findings

Deterministic o(n)-query algorithms cannot achieve better than logarithmic approximation.

New deterministic algorithms with o(n) queries and near-logarithmic approximation are developed.

Proves the non-existence of o(n)-query algorithms with logarithmic approximation for metric 1-median.

Abstract

Given an $n$-point metric space $(M,d)$, {\sc metric $1$-median} asks for a point $p\in M$ minimizing $\sum_{x\in M}\,d(p,x)$. We show that for each computable function $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ satisfying $f(n)=\omega(1)$, {\sc metric $1$-median} has a deterministic, $o(n)$-query, $o(f(n)\cdot\log n)$-approximation and nonadaptive algorithm. Previously, no deterministic $o(n)$-query $o(n)$-approximation algorithms are known for {\sc metric $1$-median}. On the negative side, we prove each deterministic $O(n)$-query algorithm for {\sc metric $1$-median} to be not $(\delta\log n)$-approximate for a sufficiently small constant $\delta>0$. We also refute the existence of deterministic $o(n)$-query $O(\log n)$-approximation algorithms.