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.

💡 Research Summary

This paper addresses the long‑standing open problem of approximating the Maximum Clique problem on geometric intersection graphs, specifically disk graphs in the plane and unit ball graphs in three dimensions. The authors present the first randomized Efficient Polynomial‑Time Approximation Scheme (EPTAS) and a deterministic Polynomial‑Time Approximation Scheme (PTAS) for these classes, and they also establish strong hardness results for related higher‑dimensional or non‑uniform radius cases.

Technical framework.
The core of the algorithmic contribution is a structural result about graphs that satisfy three conditions: (i) the VC‑dimension of the neighbourhood hypergraph is bounded by a constant d, (ii) the independence number α(G) is at least a constant fraction β of the vertex set, and (iii) the induced odd‑cycle packing number iocp(G) is at most 1 (i.e., no two vertex‑disjoint odd cycles appear as an induced subgraph). For any graph in this class, the authors devise a randomized (1‑ε)‑approximation algorithm running in time 2^{O(1/ε³)}·n^{O(1)}. The algorithm proceeds in three stages:

1. Sampling via ε‑nets. Using the Haussler–Welzl bound, a random sample S of size O(d·log(1/ε)/ε) is taken. With high probability S is contained in an optimal independent set, and removing S and its neighbours reduces the maximum degree of the remaining graph to O(d/ε).

2. Shortest odd‑cycle detection. In the reduced graph, the algorithm finds a shortest odd cycle C. If C is short, C together with its neighbourhood can be removed, leaving a bipartite graph on which the maximum independent set can be computed exactly.

3. Odd‑cycle transversal for long cycles. If C is long, the authors prove that a small odd‑cycle transversal T can be constructed from successive neighbourhoods of C. Deleting T again yields a bipartite graph, and the loss in the objective is bounded by ε.

The randomness is confined to the sampling step; a deterministic PTAS follows by enumerating all ε‑nets of the appropriate size.

Application to geometric graphs.
Bonnet et al. (SoCG 2018) showed that the complement of any disk graph cannot contain two disjoint odd cycles as an induced subgraph. The present paper supplies a new geometric proof for unit ball graphs, based on a “Kakeya motion” argument on the 2‑sphere: two needles performing such motions must intersect, which contradicts the existence of two independent odd cycles. Consequently, both disk graphs and unit ball graphs belong to the X(d,β,1) class, and the above EPTAS directly yields a (1‑ε)‑approximation for Maximum Clique (equivalently, for finding a largest subset of points of diameter ≤ 1) in time 2^{O(1/ε³)}·n^{O(1)}. This improves upon the previously best known 2‑approximation for unit ball graphs and the QPTAS/sub‑exponential algorithm for general disk graphs.

Hardness results.
The authors also prove that the situation changes dramatically when the radii are not uniform or the dimension is increased. They show:

• For 3‑dimensional ball graphs (disks of arbitrary radii), Maximum Clique is NP‑hard and does not admit any sub‑exponential‑time approximation scheme unless the Exponential Time Hypothesis (ETH) fails.
• For 4‑dimensional unit ball graphs, the problem remains NP‑hard and likewise resists any sub‑exponential‑time PTAS under ETH.

These lower bounds are obtained by reductions that exploit the fact that the forbidden induced subgraph property (no two disjoint odd cycles) no longer holds in these settings.

Significance.
The paper unifies structural graph theory (VC‑dimension, odd‑cycle packing) with geometric arguments to obtain optimal approximation schemes for a problem that had resisted exact algorithms for decades. It also delineates a clear boundary: as soon as either the dimension exceeds three or radii become non‑uniform, the problem becomes intractable even for approximation. This work therefore settles the approximability of Maximum Clique on disk and unit ball graphs, introduces a novel algorithmic paradigm based on sampling and odd‑cycle transversals, and opens new directions for studying geometric intersection graphs beyond the planar or uniform‑radius regime.