Approximating k-Connected m-Dominating Sets

Abstract

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 $v \in V {\setminus } S$ 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 $m \ge k$ we obtain the following approximation ratios. For unit disk graphs we improve the ratio $O(k \ln k)$ of Nutov (Inf Process Lett 140:30–33, 2018) to $\min \left\{ \frac{m^2}{(m-k+1)^2},k^{2/3}\right\} \cdot O(\ln ^2 k)$—this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln ^2 k)/\epsilon ^2$ when $m \ge (1+\epsilon )k$; furthermore, we obtain ratio $\min \left\{ \frac{m}{m-k+1},\sqrt{k}\right\} \cdot O(\ln ^2 k)$ for uniform weights. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ of Nutov (2018) and matches the best known ratio for unit weights of Zhang et al. (INFORMS J Comput 30(2):217–224, 2018). 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.