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.
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{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. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexp…
In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometr…
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{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. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorit…
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-ha…
A disk graph is an intersection graph of disks in the Euclidean plane, where the disks correspond to the vertices of the graph and a pair of vertices are adjacent if and only if their corresponding disks intersect. The problem of determining the time complexity of computing a maximum clique in a disk graph is a long-standing open question. The problem is known to be open even when the radii of all the disks are in the interval $[1,(1+\varepsilon)]$, where $\varepsilon>0$. However, the maximum clique problem is known to be APX-hard for the intersection graphs of many other convex objects such a…
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 …
Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs
Rutgers University（罗格斯大学）；University of Wrocław（弗罗茨瓦夫大学）；City St George’s, University of London（伦敦大学圣乔治学院）；Freie Universität Berlin（柏林自由大学）；Aalto University（阿尔托大学）；Utrecht University（乌得勒支大学）；Ben Gurion University of the Negev（内盖夫本古里安大学）
Let $F$ be a set of $n$ objects in the plane and let $G(F)$ be its intersection graph. A balanced clique-based separator of $G(F)$ is a set $S$ consisting of cliques whose removal partitions $G(F)$ into components of size at most $δn$, for some fixed constant $δ<1$. The weight of a clique-based separator is defined as $\sum_{C\in S}\log (|C|+1)$. Recently De Berg et al. (SICOMP 2020) proved that if $S$ consists of convex fat objects, then $G(F)$ admits a balanced clique-based separator of weight $O(\sqrt{n})$. We extend this result in several directions, obtaining the following results. Map gr…
We consider problems related to finding short cycles, small cliques, small independent sets, and small subgraphs in geometric intersection graphs. We obtain a plethora of new results. For example: * For the intersection graph of $n$ line segments in the plane, we give algorithms to find a 3-cycle in $O(n^{1.408})$ time, a size-3 independent set in $O(n^{1.652})$ time, a 4-clique in near-$O(n^{24/13})$ time, and a $k$-clique (or any $k$-vertex induced subgraph) in $O(n^{0.565k+O(1)})$ time for any constant $k$; we can also compute the girth in near-$O(n^{3/2})$ time. * For the intersection grap…
Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph $G$ with at most $4ω(…
We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), $P_k$-Hitting for $k\in\{3,4,5\}$, Path Deletion, Pathwidth $1$-Deletion, Component Order Connectivity, Bounded Degre…
We study the Steiner Tree problem on unit disk graphs. Given a $n$ vertex unit disk graph $G$, a subset $R\subseteq V(G)$ of $t$ vertices and a positive integer $k$, the objective is to decide if there exists a tree $T$ in $G$ that spans over all vertices of $R$ and uses at most $k$ vertices from $V\setminus R$. The vertices of $R$ are referred to as terminals and the vertices of $V(G)\setminus R$ as Steiner vertices. First, we show that the problem is NP-Hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in $n^{O(\sqrt{t+k})}$ time. We also show that the Stei…
Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new…
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and $O(m^{1/2}\log n)$ …
