Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs

TL;DR

This paper introduces faster randomized algorithms for approximating maximum cliques in disk graphs, achieving near-linear expected time for unit disks and parameterized schemes for graphs with a fixed number of radii.

Contribution

It provides the first near-linear time approximation algorithm for maximum cliques in unit disk graphs and a parameterized approximation scheme for disk graphs with multiple radii.

Findings

Approximate maximum clique in unit disk graphs in expected $ ilde{O}(n/ ext{epsilon}^2)$ time.

Parameterized approximation scheme for disk graphs with $t$ radii in expected $ ilde{O}(f(t) imes (1/ ext{epsilon})^{O(t)} imes n)$ time.

Improved algorithms using randomization and approximation for geometric intersection graphs.

Abstract

A \emph{disk graph} is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time $O(n^{7/3+ o(1)})$, where $n$ denotes the number of disks~\cite{EspenantKM23, keil_et_al:LIPIcs.SoCG.2025.63}. Moreover, for the case of disk graphs with $t$ distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time $O^*(n^{2t})$~\cite{keil_et_al:LIPIcs.SoCG.2025.63}. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: - for unit disk graphs, we give an algorithm that, with constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(n/\varepsilon^2)$; and - for disk graphs with $t$ distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$, for some (exponential) function $f(t)$.