Approximation and Inapproximability Results for Maximum Clique of Disc Graphs in High Dimensions

TL;DR

This paper investigates the maximum diameter subset problem in Euclidean space, providing approximation algorithms with near-optimal guarantees and establishing NP-hardness results that limit how closely the problem can be approximated.

Contribution

It introduces polynomial-time algorithms for approximating maximum diameter sets and proves hardness results showing the limits of approximation under P≠NP.

Findings

Polynomial-time algorithm approximates maximum diameter sets within a factor of . 

Hardness results show no approximation within a factor of . unless P=NP.

Results apply to high-dimensional Euclidean spaces.

Abstract

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean space, then for every $\epsilon>0$ there exists a polynomial time algorithm that outputs a set $B$ of size at least $|A^*|$ and of diameter at most $r(\sqrt{2}+\epsilon)$. On the hardness side, roughly speaking, we show that unless $P=NP$ for every $\epsilon>0$ it is not possible to guarantee the diameter $r(\sqrt{4/3}-\epsilon)$ for $B$ even if the algorithm is allowed to output a set of size $({95\over 94}-\epsilon)^{-1}|A^*|$.