Analysis on Theoretical Bounds for Approximating Dominating Set Problems

Connected Dominating Set is widely used as virtual backbone in wireless networks to improve network performance and optimize routing protocols. Based on special characteristics of ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacent if the distance between them is less than 1. Since every Maximal Independent Set (MIS) is a dominating set and it is easy to construct, we can firstly find an MIS and then connect it into a Connected Dominating Set (CDS). Therefore, the ratio to compare the size of an MIS with a minimum CDS becomes a theoretical upper bound for approximation algorithms to compute CDS. In our paper, with the help of Voronoi diagram and Euler's formula, we improved this upper bound, so that improved the approximations based on this relation.

[1]  Charles J. Colbourn,et al.  Unit disk graphs , 1991, Discret. Math..

[2]  Michael Segal,et al.  A simple improved distributed algorithm for minimum CDS in unit disk graphs , 2005, WiMob'2005), IEEE International Conference on Wireless And Mobile Computing, Networking And Communications, 2005..

[3]  P. Strevens.  Iii , 1985 .

[4]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[5]  Harry B. Hunt,et al.  NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs , 1998, J. Algorithms.

[6]  Peng-Jun Wan,et al.  Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[7]  Weili Wu,et al.  Improving Construction for Connected Dominating Set with Steiner Tree in Wireless Sensor Networks , 2006, J. Glob. Optim..

[8]  Gai-tai Huang,et al.  Casting the Wireless Sensor Net , 2003 .

[9]  Weili Wu,et al.  Minimum connected dominating sets and maximal independent sets in unit disk graphs , 2006, Theor. Comput. Sci..

[10]  Harry B. Hunt,et al.  Simple heuristics for unit disk graphs , 1995, Networks.

[11]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[12]  Xiuzhen Cheng,et al.  Connected Dominating Set in Sensor Networks and MANETs , 2004 .

[13]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites. , 1908 .

[14]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[15]  Nader Mohamed,et al.  Middleware: middleware challenges and approaches for wireless sensor networks , 2006, IEEE Distributed Systems Online.

[16]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .