摘要：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 â^^ 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 [Z. Nutov, 2018] and matches the best known ratio for unit weights of [Z. Zhang et al., 2018]. For unit disk graphs we improve the ratio O(k ln k) of [Z. Nutov, 2018] to min{m/(m-k),k^{2/3}} â<. O(ln² k) - this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln² k)/ε when m ⥠(1+ε)k; furthermore, we obtain ratio min{m/(m-k), â^Sk} â<. 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.
