Computing Maximum Cliques in Unit Disk Graphs

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 $O(n^{7\mathrm{/}3+o(1)})$ time. We show that the problem can be solved in $O(n\log n+n{K}^{4\mathrm{/}3+o(1)})$ 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 $O(n^{15\mathrm{/}7+o(1)})=O(n^{2.143})$ 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.