Approximation Algorithm for Minimum Weight Connected m-Fold Dominating Set

Using connected dominating set (CDS) to serve as a virtual backbone in a wireless networks can save energy and reduce interference. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. A k-connected m-fold dominating set ((k,m)-CDS) of a graph G is a node set D such that every node in V \ D has at least m neighbors in D and the subgraph of G induced by D is k-connected. Using (k,m)-CDS can tolerate the failure of min{k − 1, m− 1} nodes. In this paper, we study Minimum Weight (1, m)-CDS problem ((1, m)-MWCDS), and present an (H(δ+m)+2H(δ−1))-approximation algorithm, where δ is the maximum degree of the graph and H(·) is the Harmonic number. Notice that there is a 1.35 lnn-approximation algorithm for the (1, 1)-MWCDS problem, where n is the number of nodes in the graph. Though our constant in O(ln ·) is larger than 1.35, n is replaced by δ. Such a replacement enables us to obtain a 3.67-approximation for the connecting part of (1, m)-MWCDS problem on unit disk graphs. Keyword: m-fold dominating set, connected dominating set, non-submodular function, greedy algorithm, unit disk graph.