A Simpler Constant Factor Approximation for the k-Connected m-Domination Set Problem in Unit Disk Graph

Over years, many efforts are made for the problem of constructing quality fault-tolerant virtual backbones in wireless network. In case that a wireless network consists of physically equivalent nodes, e.g. with the same communication range, unit disk graph (UDG) is widely used to abstract the wireless network and the problem is formulated as the minimum k-connected mdominating set problem on the UDG. So far, most results are focused on designing a constant factor approximation algorithm for this NP-hard problem under two positive integers k and m satisfying m ≥ k ≥ 1 and k ≤ 3. Very recently, Shi et. al. and Fukunaga separately introduced constant factor approximation algorithms for the problem with m ≥ k ≥ 1. However, we found the structures of the algorithms are extremely complicated, and thus it would be difficult to implement and use them in practice. Motivated by such observation, this paper introduces a novel approximation algorithm for the problem with m ≥ k ≥ 1. This algorithm is based on our new technique which first computes a 1-connected m-dominating set D and repeatedly (a) decomposes D into an i-connected block tree, with i = 2, 3, ··· , k, and (b) use this graph structure to improve the connectivity of D, until D becomes k-connected. We provide a rigorous theoretical analysis to prove that the proposed algorithm is correct and its approximation ratio is a constant. We compare the structure of our algorithm against the existing ones and show our algorithm is much simpler to understand and implement.