Approximation Algorithms for Unit Disk Graphs Erik Jan van Leeuwen institute of information and computing sciences, utrecht university technical report UU-CS-2004-066 www.cs.uu.nl Approximation Algorithms for Unit Disk Graphs∗ Erik Jan van Leeuwen Institute for Information and Computing Sciences, Utrecht University, Padualaan 14, 3584 CH Utrecht, the Netherlands. E-mail: [email protected] Abstract Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks. We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in any width 1 slab. If the thickness of a graph is bounded, then the considered problems can be solved in polynomial time. We prove this both indirectly by presenting a relation between unit disk graphs of bounded thickness and the pathwidth of such graphs, and directly by giving dynamic programming algorithms. This result implies that the problems are fixed-parameter tractable in the thickness. We then consider unit disk graphs of bounded density. The density of a graph is the number of disk centers in any 1-by-1 box. We present a new approximation scheme for the considered problems, which uses the bounded thickness results mentioned above and the so called shifting technique. We show that the scheme is an asymptotic FPTAS and that this result is optimal, in the sense that no FPTAS can exist (unless P=NP). The scheme for Minimum Connected Dominating Set is the first FPTAS∞ for this problem. The analysis that is applied can also be used to improve existing results, which among others implies the existence of an FPTAS∞ for MCDS on planar graphs. 1 Introduction Mobile ad hoc networks are the next generation in communication networks. Devices can enter or leave the network anytime, devices can be mobile, and no centralized control points or base stations are necessary. This flexibility makes mobile ad hoc networks very interesting for the consumer and military market, but also makes them more complex than many existing wireless networks (such as GSM). The distinct properties of mobile ad hoc networks have given rise to various new problems and challenges. These can be very practical (such as message routing) and theoretical. This paper focusses on the more theoretical aspects of mobile ad hoc networks and con- siders a graph model for such networks. As will be shown, a mobile ad hoc network can be naturally modeled as a so called (unit) disk graph. Each node in such a graph has a disk around it containing all points reachable by that node. The intersections of these disks then ∗This research was partially supported by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorical Optimisation). 1 2 1 INTRODUCTION determine the edges of the graph. As these graphs have a nice geometric interpretation, we may hope that classical graph theoretic problems that are relevant for mobile ad hoc networks (such as Maximum Independent Set and Minimum (Connected) Dominating Set) are easier to solve or approximate than they are on general graphs. We show that by making some realistic assumptions about this geometric interpretation, various new properties and algorithms can be found, which improve on known results. 1.1 Preliminaries Unit disk graphs are a special kind of (geometric) intersection graphs. Hence, we first define these graphs. Definition 1.1 Let S be a set of geometric objects. Then the graph G = (V, E), where each vertex corresponds to an object in S and two vertices are connected by an edge if and only if the two corresponding objects intersect, is called an intersection graph. The graph G is said to be realized by S. In this definition, tangent objects are assumed to intersect. We can now formally define (unit) 2 disk graphs. Denote by ci ∈ R the center and by ri the radius of a disk Di. Definition 1.2 A graph G is a disk graph if and only if there exists a set of disks D = {Di | i = 1, . , n}, such that G is the intersection graph of D. The set of disks is called a disk representation of G. A disk graph can be given without its disk representation. In this case, it is assumed each vertex knows the vertices adjacent to it. However, knowing a disk representation can help to more efficiently solve problems on disk graphs. Definition 1.3 A graph G is a unit disk graph if and only if G is a disk graph and the radii of a set of disks realizing G are equal. 1 Usually the common radius is 1, but often it is assumed to be 2 . Note that any common radius can be obtained by scaling D appropriately. We use the following notation. Notation 1.4 Let G = (V, E) be a disk graph with disk representation D = {Di | i = 1, . , n}. For a vertex v ∈ V , we denote the corresponding disk in D by Dv, its center by cv, and its radius by rv. For a disk center ci of some disk in D, we denote the vertex v ∈ V corresponding to that disk by v(ci). Observe that (unit) disk graphs are a good model for mobile ad hoc networks. Each node of the network corresponds to a disk center and the transmission range of a node corresponds to the radius of the disk. In a unit disk graph, all nodes are assumed to have the same transmission range. We also define the various approximation scheme types considered in this paper. Definition 1.5 Let P be a maximization (minimization) problem. Then an algorithm A is a • polynomial-time approximation scheme (PTAS) for P if and only if for any instance x of P and for any (fixed) > 0, A(x, ) runs in time polynomial in |x| and delivers a feasible solution with value SOLx,, such that SOLx, ≥ (1 − )OP Tx (SOLx, ≤ (1 + )OP Tx). 1.2 Problem definitions 3 • fully polynomial-time approximation scheme (FPTAS) for P if and only if for any 1 instance x of P and for any > 0, A(x, ) runs in time polynomial in |x| and and delivers a feasible solution with value SOLx,, such that SOLx, ≥ (1 − )OP Tx (SOLx, ≤ (1 + )OP Tx). • asymptotic fully polynomial-time approximation scheme (FPTAS∞) for P if and only if for any instance x of P and for any > 0, A(x, ) runs in time polynomial in 1 |x| and and delivers a feasible solution with value SOLx,. If |x| > c, then also SOLx, ≥ (1 − )OP Tx (SOLx, ≤ (1 + )OP Tx). Finally, we define fixed-parameter tractability, following Downey and Fellows [19]. Definition 1.6 Let hx, ki be an instance of a parameterized problem P , with parameter k ∈ N. Then P is fixed-parameter tractable (FPT) if and only if there exists an algorithm that delivers a feasible solution for hx, ki in running time O(f(k) poly(|x|)) for all x, where f is an arbitrary function and poly(|x|) is an arbitrary polynomial in |x|. Note that a fixed-parameter tractable problem can be solved in polynomial time, given any fixed k. 1.2 Problem definitions We consider various classical optimization problems on graphs, relevant to (unit) disk graph models of mobile ad hoc networks. Definition 1.7 Let G = (V, E) be a graph. A set S ⊆ V is an independent set if and only if there are no u, v ∈ S, such that (u, v) ∈ E. A set S ⊆ V is a vertex cover if and only if for each (u, v) ∈ E it holds that u ∈ S or v ∈ S. Observe that an independent set is the complement of a vertex cover (and vice versa) [22]. Furthermore, we are usually looking for a maximum independent set and a minimum vertex cover. An independent set is maximum if and only if there is no independent set of greater size. A similar definition holds for minimum vertex cover. In the context of mobile ad hoc networks, an independent set of a (unit) disk graph can be seen as a set of nodes that can transmit simultaneously without signal interferences. Vertex covers are mostly interesting from a theoretical point of view. Definition 1.8 Let G = (V, E) be a graph. A set S ⊆ V is a dominating set if and only if for each vertex v either v ∈ S or there exists a vertex u ∈ S for which (u, v) ∈ E. Definition 1.9 Let G = (V, E) be a graph. A set S ⊆ V is a connected dominating set if and only if S is a dominating set and the subgraph of G induced by S (G[S] = (S, (S ×S)∩E)) is connected. A dominating set in a mobile ad hoc network can be seen as a set of emergency transmitters capable of reaching every node in the network, or as central nodes in node clusters. A connected dominating set can be used as a backbone for easier and faster communications. The problem is to find a minimum (connected) dominating set. 4 1 INTRODUCTION 1.3 Previous work All problems mentioned above are NP-hard for general graphs (see Garey and Johnson [22]). Since (unit) disk graphs are a restricted class of graphs with a nice geometric interpretation, one might hope that these problems are better solvable. Unfortunately, Clark, Colbourn, and Johnson [16] proved these problems to be NP-hard on (unit) disk graphs as well.