The Maximum Clique Problem in a Disk Graph Made Easy
J. Mark Keil, Debajyoti Mondal
TL;DR
This paper addresses the long-standing open problem of finding a maximum clique in disk graphs with arbitrary radii by introducing a slab-based geometric framework to study adjacencies. The authors develop a modular approach that yields a polynomial-time algorithm for disk graphs with k distinct radii when k is fixed, and extend the method to rectangular range queries and to ball graphs under restricted plane configurations, achieving time bounds that scale as O(n^{2k}(f(n)+n^2)) and O(n^{2rk}(f(n)+n^2 r)) respectively. Key contributions include a concrete O(n^{2k} (f(n)+n^2))-time algorithm for disk graphs with k radii, an O(n^5 log n) preprocessing for maximum clique across all axis-aligned rectangles in unit-disk arrangements, and a generalized O(n^{2rk} (f(n)+n^2 r))-time algorithm for ball graphs on r planes. These results provide new tractable regimes for maximum clique in geometric intersection graphs and introduce a versatile slab-based technique that contrasts with prior lens-based methods, with potential impact on related range queries and higher-dimensional variants.
Abstract
A disk graph is an intersection graph of disks in $\mathbb{R}^2$. Determining the computational complexity of finding a maximum clique in a disk graph is a long-standing open problem. In 1990, Clark, Colbourn, and Johnson gave a polynomial-time algorithm for computing a maximum clique in a unit disk graph. However, finding a maximum clique when disks are of arbitrary size is widely believed to be a challenging open problem. The problem is open even if we restrict the disks to have at most two different sizes of radii, or restrict the radii to be within $[1,1+\varepsilon]$ for some $ε>0$. In this paper, we provide a new perspective to examine adjacencies in a disk graph that helps obtain the following results. - We design an $O(2^k n^{2k} poly(n))$-time algorithm to find a maximum clique in a $n$-vertex disk graph with $k$ different sizes of radii. This is polynomial for every fixed $k$, and thus settles the open question for the case when $k=2$. - Given a set of $n$ unit disks, we show how to compute a maximum clique inside each possible axis-aligned rectangle determined by the disk centers in $O(n^5\log n)$-time. This is at least a factor of $n^{4/3}$ faster than applying the fastest known algorithm for finding a maximum clique in a unit disk graph for each rectangle independently. - We give an $O(2^kn^{2rk} poly(n,r))$-time algorithm to find a maximum clique in a $n$-vertex ball graph with $k$ different sizes of radii where the ball centers lie on $r$ parallel planes. This is polynomial for every fixed $k$ and $r$, and thus contrasts the previously known NP-hardness result for finding a maximum clique in an arbitrary ball graph.