Approximating k-Connected m-Dominating Sets

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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-Connectedm-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 unit disk graphs we improve the ratio O(kln k) of Nutov (Inf Process Lett 140:30–33, 2018) to min{m2(m-k+1)2,k2/3}·O(ln2k)—this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln 2k) / ϵ2 when m≥ (1 + ϵ) k; furthermore, we obtain ratio min{mm-k+1,k}·O(ln2k) for uniform weights. For general graphs our ratio O(kln n) improves the previous best ratio O(k2ln 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 Subsetk-Connectivity problem when the set of terminals is an m-dominating set.

Original languageEnglish
Article number6
Pages (from-to)1511-1525
Number of pages15
Issue number6
Early online date16 Feb 2022
StatePublished - Jun 2022


  • Approximation algorithms
  • Subset k-connectivity
  • k-connected graph
  • m-dominating set


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