Approximating k-Connected m-Dominating Sets in Disk Graphs

Abstract

This paper considers dominating set problems in a disk graphs, which is a generalization of unit disk graphs extensively used to analyze homogeneous sensor or wireless networks. When considering heterogeneous networks, it is useful to consider disk graphs that contain disks with different radii. Given graph $G=(V,E)$, set $D\subseteq V$ is a $(k,m)$ -connected dominating set for G if every node in V is either in D or has at least m neighbors in D and the induced subgraph $G[D]$ is also k -connected. Many approximation algorithms are known for this problem in unit disk graphs. We prove various properties for disk graphs so that these algorithms can be generalized to disk graphs. Namely, we show that a $\displaystyle \min\left\{\frac{m}{m-k},\sqrt{k}\right\}\cdot O\left(\ln^{2}k\right)$ - approximation algorithm of Nutov works in this setting. We also present a PTAS for finding a $(1+\epsilon)$ -approximate solution to the m -dominating set problem in disk graphs that runs in time $n^{O(m/\epsilon)}$