EPTAS for Max Clique on Disks and Unit Balls

Abstract

We propose a polynomial-time algorithm which takes as input a finite set of points of R 3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane.Almost three decades ago, an elegant polynomial-time algorithm was found for 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. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzążewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VCdimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.

Computing a maximum clique in geometric superclasses of disk graphs

“…We answer in Section 4 the 2-dimensional version of the question asked by Bonamy et al [4]: We present a polynomial time algorithm for computing a maximum clique in a geometric superclass of interval graphs and unit disk graphs.…”

“…However in 2017, Bonnet et al, found a subexponential algorithm and a quasi polynomial time approximation scheme (QPTAS) for computing a maximum clique on disk graphs [5]. The following year, Bonamy et al extended the result to unit ball graphs, and gave a randomised EPTAS for both settings [4]. The current state-of-the-art about the complexity of computing a maximum clique in d-ball graphs is summarised in Table 1.…”

“…The current state-of-the-art about the complexity of computing a maximum clique in d-ball graphs is summarised in Table 1. polynomial [7] Unknown but EPTAS [4, 5] d = 3 Unknown but EPTAS [4] NP-hard [4]…”

“…NP-hard [4] NP-hard [4] Table 1. Complexity of computing a maximum clique on d-ball graphs Bonamy et al show that the existence of an EPTAS follows from the fact that: For any graph G that is a disk graph or a unit ball graph, the disjoint union of two odd cycles is a forbidden induced subgraph in the complement of G. Surprisingly, the proofs for disk graphs on one hand and unit ball graphs on the other hand are not related.…”

A Tutorial on Clique Problems in Communications and Signal Processing

“…This is accomplished by casting the maximum weight clique problem as an integer program and designing a local search method that avoids entrapment in local optima thanks to a couple of perturbations. Likewise, the authors in [76] propose a polynomial time algorithm so as to discover the maximum clique in unit ball graphs.…”