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

2025-12-10

AI Summary

This paper studies efficient approximation algorithms for the maximum clique problem on disk graphs. For unit disk graphs, we present the first $(1-varepsilon)$-approximation algorithm with expected time complexity $ ilde{O}(n/varepsilon^2)$, achieving near-linear runtime. For disk graphs with $t$ distinct radii, we design the first $t$-parameterized approximation algorithm, running in $ ilde{O}(f(t) cdot (1/varepsilon)^{O(t)} cdot n)$ time—substantially improving upon the previous exact $O^*(n^{2t})$ algorithm. Our approach integrates randomized sampling, geometric hierarchical partitioning, local clique enumeration, and parameterized pruning. These techniques collectively break the long-standing runtime barrier for approximating maximum cliques in unit disk graphs and, for the first time, explicitly characterize the dependence of approximation efficiency on the number $t$ of distinct radii—establishing a precise parametric trade-off between geometric diversity and computational tractability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $ ilde{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 $ ilde{O}(f(t)cdot (1/varepsilon)^{O(t)} cdot n)$.

Problem

• Finding maximum cliques in disk graphs efficiently
• Improving time complexity for unit disk graph approximations
• Developing parameterized approximation for multiple radius disks

Innovation

• Randomized algorithm for near-linear time unit disk graph approximation
• Parameterized approximation scheme for disk graphs with distinct radii
• Constant success probability for approximate maximum clique solutions