Approximation and Inapproximability Results for Maximum Clique of Disc Graphs in High Dimensions

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean space, then for every $\epsilon>0$ there exists a polynomial time algorithm that outputs a set $B$ of size at least $|A^*|$ and of diameter at most $r(\sqrt{2}+\epsilon)$. On the hardness side, roughly speaking, we show that unless $P=NP$ for every $\epsilon>0$ it is not possible to guarantee the diameter $r(\sqrt{4/3}-\epsilon)$ for $B$ even if the algorithm is allowed to output a set of size $({95\over 94}-\epsilon)^{-1}|A^*|$.Comment: Final versio

EPTAS for Max Clique on Disks and Unit Balls

We propose a polynomial-time algorithm which takes as input a finite set of points of $\mathbb R^3$ and compute, 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. 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, Rz\k{a}\.{z}ewski, 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 VC-dimension, and large 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 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.Comment: 19 pages, 3 figure

Approximation and Inapproximability Results for Maximum Clique of Disc Graphs in High Dimensions

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular we prove that if A is the largest subset of diameter r of n points in the Euclidean space, then for every ǫ> 0 there exists a polynomial time algorithm that outputs a set B of size at least |A | and of diameter at most r ( √ 2 + ǫ). On the hardness side roughly speaking we show that unless P = NP for every ǫ> 0 it is not possible to guarantee the diameter r ( √ 4/3 − ǫ) for B.