EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

TL;DR

This paper develops efficient approximation schemes and subexponential algorithms for the Maximum Clique problem on disk and unit ball graphs, leveraging structural properties and complexity results.

Contribution

It introduces a simple QPTAS, a subexponential algorithm, and a randomized EPTAS for Maximum Clique on disk and unit ball graphs, based on new structural insights.

Findings

A disjoint union of two odd cycles is not the complement of a disk or unit ball graph.

A subexponential algorithm runs in time 2^{~O(n^{2/3})} for Maximum Clique on these graphs.

A randomized EPTAS is achieved for graphs with certain structural properties.

Abstract

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for \textsc{Max Clique} on disk and unit ball graphs. \textsc{Max Clique} on unit ball graphs is equivalent to finding, given a collection of points in $\mathbb R^3$, a maximum subset of points with diameter at most some fixed value. In stark contrast, \textsc{Maximum Clique} on ball graphs and unit $4$-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.