Performance Guaranteed Approximation Algorithm for Minimum $k$-Connected $m$-Fold Dominating Set

To achieve an efficient routing in a wireless sensor network, connected dominating set (CDS) is used as virtual backbone. A fault-tolerant virtual backbone can be modeled as a (k,m)-CDS. For a connected graph G = (V,E) and two fixed integers k and m, a node set C ⊆ V is a (k,m)-CDS of G if every node in V \C has at least m neighbors in C, and the subgraph of G induced by C is k-connected. Previous to this work, approximation algorithms with guaranteed performance ratio in a general graph were know only for k ≤ 3. This paper makes a significant progress by presenting a (2k− 1)α0 approximation algorithm for general k and m with m ≥ k, where α0 is the performance ratio for the minimum CDS problem. Using currently best known ratio for α0, our algorithm has performance ratio O(ln∆), where ∆ is the maximum degree of the graph. Keyword: wireless sensor network; virtual backbone; connected dominating set; fault-tolerance; approximation algorithm.