QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

TL;DR

This paper introduces a quasi-polynomial time approximation scheme and a subexponential algorithm for the Maximum Clique problem on disk graphs, revealing structural insights and contrasting complexity with other geometric intersection graphs.

Contribution

It provides the first QPTAS and subexponential algorithm for Maximum Clique on disk graphs, based on a novel structural characterization of their complements.

Findings

First QPTAS for Maximum Clique on disk graphs

Subexponential algorithm with time 2^{~O(n^{2/3})}

Hardness results for ellipses and triangles intersection graphs

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 the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant approximation which is not attainable even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.