A \emph{disk graph} is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time $O(n^{7/3+ o(1)})$, where $n$ denotes the number of disks~\cite{EspenantKM23, keil_et_al:LIPIcs.SoCG.2025.63}. Moreover, for the case of disk graphs with $t$ distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time $O^*(n^{2t})$~\cite{keil_et_al:LIPIcs.SoCG.2025.63}. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization:
(i) for unit disk graphs, we give an algorithm that, with constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(n/\varepsilon^2)$; and
(ii) for disk graphs with $t$ distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$.
A disk graph is the intersection graph of closed disks in the plane, where the vertices are the disks, and two vertices are connected by an edge if and only if the two corresponding disks intersect. A unit disk graph is a disk graph in which all disks have the same radius, which can be assumed to be 1. Disk graphs and unit disk graphs are probably the most extensively studied classes of geometric intersection graphs [4,5,7,14,19,29]. In particular, they are popular models of wireless communication networks, where each disk represents a wireless station, and the respective radii encode the transmission ranges. Theoretically, unit disk graphs and disk graphs are of special interest, because they generalize the familiar class of planar graphs. Indeed, the well-known circle-packing theorem (also called the the Koebe-Andreev-Thurston theorem [26]) shows that every planar graph is a disk graph. Unlike planar graphs, disk graphs (or unit disk graphs) can be dense, with potentially Θ(n 2 ) edges. Thus, algorithms on disk graphs often use an implicit representation (of the coordinates and radii of the disks) and derive edges on-demand.
Due to the special structure of disk graphs, it is an active research direction to study efficient approximation algorithms for classical NP-hard problems on disk graphs or unit disk graphs, such as maximum independent set [11,22] and minimum dominating set [20,28]. In this paper, we study the maximum clique problem on unit disk graphs and disk graphs. For a given graph G = (V, E), the problem asks for a vertex set S ⊆ V of maximum cardinality such that the induced subgraph on S is a clique. The maximum clique problem on general graphs is a classical NP-hard problem and is also hard to approximate [17]. However, maximum clique can be solved in polynomial time on unit disk graphs and the problem has been actively studied recently. We first review prior work on this problem and then present our new results. Related work. Clark, Colbourn, and Johnson [14] gave an elegant O(n 4.5 )-time algorithm for finding a maximum clique in a unit disk graph. The algorithm guesses in quadratic time the pair of most distant (diameter) disk centers in a maximum clique and considers the subgraph induced by all the disks whose centers are at most as distant from both candidate diameter centers. The induced graph is co-bipartite, hence computing a maximum clique is equivalent to finding a maximum independent set in the complement bipartite graph. This in turn reduces to finding a maximum matching. By reducing the necessary number of pairs of centers to search for and/or reducing the running time for finding a maximum independent set, the running time has been successively improved to O(n 3.5 log n) by Breu [6], O(n 3 log n) by Eppstein [15], O(n 2.5 log n) by Espenant, Keil and Mondal [16] and, most recently, to O(n 7/3+o (1) ) by Chan [10] as noted in Keil and Mondal [24]. In particular, the latter result is based on the clever divide and conquer approach by Espenant, Keil and Mondal [16], which considers only a linear number of center pairs, and on the observation that, for each such pair, one can use a small bi-clique cover of size 1 Õ(n 4/3 ) of the resulting bipartite graph to compute a maximum matching in time almost linear to the size of cover. Very recently, Tkachenko and Wang [29] gave an algorithm that runs in time O(n log n + nK 4/3+o (1) ), where K is the size of the maximum clique; this is an improvement to the above bound by Chan [10] when
For general disk graphs, the complexity of computing the maximum clique has been a long-standing open problem [3,16,19]. Recently, Keil and Mondal [24] considered the case where there are t distinct disk radii and gave an algorithm that runs in polynomial time for every constant t. In particular, the running time is O(2 t n 2t (f (n) + n 2 )), where f (n) is the time to compute a maximum matching in an n-vertex bipartite graph. A maximum matching can be computed in time Õ(m) using the near-linear time max-flow algorithm in the recent breakthrough in [12], where m is the number of edges in the bipartite graph, or in time Õ(n 2 ) by a combinatorial algorithm [13]. A 2-approximation can be easily found in linear time [9] by computing a stabbing set of four points [3]. Further, the problem admits an EPTAS 2 with a running time of 2 Õ(1/ε 3 ) n O (1) and an exact sub-exponential time algorithm [4] (both of which also generalize to unit ball graphs).
Finally, the problem has also been studied for intersection graphs of other classes of objects in the plane and has been shown to be NP-hard for rays [8], strings [25], ellipses and triangles [3], and for sets of both rectangles and unit disks [5].
Our contribution. The above recent exciting results and the absence of any non-trivial lower bounds for the problem leave some natural open questions.
For improving the currently best O(n 7/3+o (1) )-time bound for unit disk graphs, any algorithm that follows the clas
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