Constant-approximation algorithms for highly connected multi-dominating sets in unit disk graphs

Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((k,m)-CDS) is defined as a subset S of V such that each node in V \ S has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (k,m)-CDS problem is to find a minimum weight (k,m)-CDS in a given node-weighted graph. The problem is called the unweighted (k,m)-CDS problem if the objective is to minimize the cardinality of a (k,m)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. However, constant-approximation algorithms are known only for k ≤ 3 in the unweighted (k,m)-CDS problem, and for (k,m) = (1, 1) in the weighted (k,m)-CDS problem. In this paper, we consider the case in which m ≥ k, and we present a simple O(5k!)-approximation algorithm for the unweighted (k,m)-CDS problem, and a primal-dual O(k log k)-approximation algorithm for the weighted (k,m)-CDS problem. Both algorithms achieve constant approximation factors when k is a fixed