Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

Channel assignment problems in the time, frequency and code domains have thus far been studied separately. Exploiting the similarity of constraints that characterize assignments within and across these domains, we introduce the first unified framework for the study of assignment problems. Our framework identifies eleven atomic constraints underlying most current and potential assignment problems, and characterizes a problem as a combination of these constraints. Based on this framework, we present a unified algorithm for efficient (T/F/C)DMA channel assignments to network nodes or to inter-nodal links in a (multihop) wireless network. The algorithm is parametrized to allow for tradeoff-selectable use as three different variants called RAND, MNF, and PMNF. We provide comprehensive theoretical analysis characterizing the worst-case performance of our algorithm for several classes of problems. In particular, we show that the assignments produced by the PMNF variant are proportional to the thickness of the network. For most typical multihop networks, the thickness can be bounded by a small constant, and hence this represents a significant theoretical result. We also experimentally study the relative performance of the variants for one node and one link assignment problem. We observe that the PMNF variant performs the best, and that a large percentage of unidirectional links is detrimental to the performance in general.

We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of -precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for -precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...

An L(2; 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of all nonnegative integers such that jf(x) 0 f(y)j 2 if d(x; y) = 1 and jf(x) 0 f(y)j 1 if d(x; y) = 2. The L(2; 1)-labeling number (G) of G is the smallest number k such that G has a L(2; 1)-labeling with maxff(v) : v 2 V (G)g = k. In this paper, we give exact formulas of (G[H) and (G+H). We also prove that (G) 1 2 +1 for any graph G of maximum degree 1. For OSF-chordal graphs, the upper bound can be reduced to (G) 21+ 1. For SF-chordal graphs, the upper bound can be reduced to (G) 1+ 2Ø(G) 0 2. Finally, we present a polynomial time algorithm to determine (T ) for a tree T . Keywords. L(2; 1)-labeling, T -coloring, union, join, chordal graph, perfect graph, tree, bipartite matching, algorithm 1 Introduction The channel assignment problem is to assign a channel (nonnegative integer) to each radio transmitter so that interfering transmitters are assigned channels whose separation is not in...

Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NP-hard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NP-hard. We conjecture that K-sphericity is NP-hard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...

A cellular network is generally modeled as a subgraph of the triangular lattice. In the static frequency assignment problem, each vertex of the graph is a base station in the network, and has associated with it an integer weight that represents the number of calls that must be served at the vertex by assigning distinct frequencies per call. The edges of the graph model interference constraints for frequencies assigned to neighboring stations. The static frequency assignment problem can be abstracted as a graph multicoloring problem. We describe an efficient algorithm to optimally multicolor any weighted even or odd length cycle representing a cellular network. This result is further extended to any outerplanar graph. For the problem of multicoloring an arbitrary connected subgraph of the triangular lattice, we demonstrate an approximation algorithm which guarantees that no more than 4=3 times the minimum number of required colors are used. Further, we show that this algorithm can be im...

Abstract — Game theory is a set of tools developed to model interactions between agents with conflicting interests, and is thus well-suited to address some problems in communications systems. In this paper we present some of the basic concepts of game theory and show why it is an appropriate tool for analyzing some communication problems and providing insights into how communication systems should be designed. We then provided a detailed example in which game theory is applied to the power control problem in a CDMA-like system. I.

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In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark, Colbourn and Johnson (1990) it is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k 3. Furthermore, a 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques, and it is pointed out how this technique can be applied to more general classes of disk graphs. Contents 1 Introduction 1 2 Preliminaries 2 3 The UD k-Colorability Problem 5 3.1 The Construction of the Auxiliary Graphs : : : : : : : : : : : : : : : : : : : : : : 6 3.2 The Embedding of the UD Graph : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2.1 The Embedding of the Given Graph : : : : : : : : : : : : : : : : : : : : : 8 3.2.2 The Embedding ...

Channel assignment problems in the time, frequency and code domains have hitherto been studied separately. Exploiting the similarity of constraints that characterize assignments within and across these domains, we introduce the first unified framework for the study of assignment problems. Our framework identifies eleven atomic constraints underlying most current and potential assignment problems, and characterizes a problem as a combination of these constraints. Based on this framework, we present a unified algorithm for efficient (T/F/C)DMA channel assignments to network nodes or to inter-nodal links in a (multihop) wireless network. The algorithm is parametrized to allow for use as three different variants - RAND, MNF, and PMNF. We provide comprehensive theoretical analysis characterizing the worst-case performance of our algorithm for several classes of problems. In particular, we show that the assignments produced by the PMNF variant are proportional to the thickness of the network...