Approximation Algorithms for Unit Disk Graphs

• Published in: Graph-Theoretic Concepts in Computer Science pp. 351 - 361
• Main Author: van Leeuwen, Erik Jan
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Bounded Density; Computer science; control theory; systems; Disk Center; Exact sciences and technology; Information retrieval. Graph; Path Decomposition; Planar Graph; Theoretical computing; Unit Disk Graph; Dominating set; Computer theory; Polynomial approximation; Planar graph; Independent graph; Graph theory; Distributed system; Approximation algorithm; Modeling; Polynomial time; Set covering; Wireless network; Asymptotic approximation; Graph covering; Ad hoc network; Mobile computing
• ISBN: 3540310002; 9783540310006
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11604686_31

We consider several graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set) relevant to mobile ad hoc networks. We propose two new notions: thickness and density. If the thickness of a unit disk graph is bounded, then the mentioned problems can be solved in polynomial time. For unit disk graphs of bounded density, we present a new asymptotic fully-polynomial approximation scheme for the considered problems. The scheme for Minimum Connected Dominating Set is the first Baker-like asymptotic FPTAS for this problem. By adapting the proof, it implies e.g. an asymptotic FPTAS for Minimum Connected Dominating Set on planar graphs.