Approximating k-connected m-dominating sets

A subset S of nodes in a graph G is a k-connected m-dominating set ((k, m)-cds) if the subgraph G[S] induced by S is k-connected and every v ∈ V \ S has at least m neighbors in S. In the k-Connected m-Dominating Set ((k, m)-CDS) problem the goal is to find a minimum weight (k, m)-cds in a node-weighted graph. For m ≥ k we obtain the following approximation ratios. For general graphs our ratio O(k ln n) improves the previous best ratio O(k² ln n) of [26] and matches the best known ratio for unit weights of [34]. For unit disk graphs we improve the ratio O(k ln k) of [26] to min { _m^m₋k, k²/³ · O(ln² k) – this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln² k)/ when m ≥ (1 + )k; furthermore, we obtain ratio min {_m^m₋k, ^√k · O(ln² k) for uniform weights. These results are obtained by showing the same ratios for the Subset k-Connectivity problem when the set of terminals is an m-dominating set.