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

TL;DR

This work advances the maximum clique problem in disk graphs by introducing randomized, near-linear-time approximations for unit disk graphs and a parameterized approximation scheme for disk graphs with t radii. It leverages reductions to co-bipartite graphs and efficient approximate independent set and matching procedures, complemented by geometry-based sampling and data-structure acceleration. The results include a (1−ε)-approximation in $\tilde{O}(n/ε^2)$ time for unit disks and an EPAS with runtime $\tilde{O}(f(t)\cdot (1/ε)^{O(t)}\cdot n)$ for t-radius disk graphs, marking a substantial improvement over prior bounds for fixed ε or fixed t. Together, these contributions offer practical, scalable approaches for dense geometric graphs in wireless and computational geometry applications.

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: (i) 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 (ii) 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 $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$.

Figures (3)

• Figure 1: The lens $D(c,r) \cap D(c',r')$ of two intersecting disks $D(c,r), D(c',r')$.
• Figure 2: A cell $C$ (the central cell) and its extension $C^+$, where each cell in $C^+$ has a diameter 2. All points (both solid and hollow) belong to $P$, while the hollow points also belong to $P_C$.
• Figure 3: The cell $C$ is represented by the light-blue colored cell with $p_1, p_2\in C$. The thick black line denotes the line segment $s=\overline{p_2 x}$ of length $2$, passing through $p_1$. The dotted disk represents $D(x,2)$, while the dash-dotted disk represents $D(p_2,2)$. The yellow-shaded region corresponds to the lens $L = D(x,2) \cap D(p_2,2)$, formed along the axis $s$.

Theorems & Definitions (24)

• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Claim 5
• proof