Computing Maximum Cliques in Unit Disk Graphs

TL;DR

This paper presents improved algorithms for finding maximum cliques in unit disk graphs, including faster solutions when the maximum clique size is small or when points are in convex position, with probabilistic and deterministic variants.

Contribution

The authors develop faster algorithms for maximum clique in unit disk graphs, especially for specific cases like convex position and known clique points, improving upon previous methods.

Findings

New algorithm with $O(n ext{log} n + n K^{4/3+o(1)})$ time complexity.

Probabilistic algorithm for convex position points with $O(n^{15/7+o(1)})$ time.

Deterministic $O(n^2 ext{log} n)$ algorithm for known maximum clique point in convex position.

Abstract

Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum clique in $G(P)$. The previously best algorithm for the problem runs in $O(n^{7/3+o(1)})$ time. We show that the problem can be solved in $O(n \log n + n K^{4/3+o(1)})$ time, where $K$ is the maximum clique size. The algorithm is faster than the previous one when $K=o(n)$. In addition, if $P$ is in convex position, we give a randomized algorithm that runs in $O(n^{15/7+o(1)})= O(n^{2.143})$ worst-case time and the algorithm can compute a maximum clique with high probability. For points in convex position, one special case we solve is when a point in the maximum clique is given; we present an $O(n^2\log n)$ time (deterministic) algorithm for this special case.