Computing Minimum <i>k</i>-Connected <i>m</i>-Fold Dominating Set in General Graphs

Zhang, Zhao; Zhou, Jiao; Tang, Shaojie; Huang, Xiaohui; Du, Dingzhu

doi:10.1287/ijoc.2017.0776

Cited by 12 publications

(6 citation statements)

References 24 publications

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“…For general graphs our ratio O(k ln n) improves the previous ratio O(k 2 ln n) of [9] and matches the best known ratio for unit weights of [11]. For unit disc graphs our ratio min k m−k , k 2/3 · O(ln 2 k) improves the previous best ratio O(k ln k) of [9]; this is the first sublinear ratio for the problem, and for any constant ǫ > 0 and m = k(1 + ǫ) the first polylogarithmic ratio O(ln 2 k)/ǫ.…”

Section: Introductionsupporting

confidence: 80%

“…For m ≥ k we obtain the following approximation ratios. For general graphs our ratio O(k ln n) improves the previous best ratio O(k 2 ln n) of [9] and matches the best known ratio for unit weights of [11]. For unit disc graphs we improve the ratio O(k ln k) of [9] to min m m−k , k 2/3 · O(ln 2 k) -this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln 2 k)/ǫ when m ≥ (1 + ǫ)k; furthermore, we obtain ratio min m m−k , √ k ·O(ln 2 k) for uniform weights.…”

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confidence: 83%

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Improved approximation algorithms for k-connected m-dominating set problems

Nutov

2018

Information Processing Letters

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A subset S of nodes in a graph G is a k-connected mdominating 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(k 2 ln n) of [9] and matches the best known ratio for unit weights of [11]. For unit disc graphs we improve the ratio O(k ln k) of [9] to min m m−k , k 2/3 · O(ln 2 k) -this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln 2 k)/ǫ when m ≥ (1 + ǫ)k; furthermore, we obtain ratio min m m−k , √ k ·O(ln 2 k) for uniform weights. These results are obtained by showing the same ratios for the Subset k-Connectivity problem when the set T of terminals is an m-dominating set with m ≥ k.

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Section: Introductionsupporting

confidence: 80%

supporting

confidence: 83%

Improved approximation algorithms for k-connected m-dominating set problems

Nutov

2018

Information Processing Letters

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“…For m ≥ k we obtain the following approximation ratios. For general graphs our ratio O(k ln n) improves the previous best ratio O(k 2 ln n) of [26] and matches the best known ratio for unit weights of [34]. For unit disk graphs we improve the ratiothis is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln 2 k)/ when m ≥ (1 + )k; furthermore, we obtain ratio min m m−k , √ k • O(ln 2 k) for uniform weights.…”

supporting

confidence: 77%

supporting

confidence: 73%

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Approximating k-Connected m-Dominating Sets

Nutov

2022

Algorithmica

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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-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(k 2 ln n) of [26] and matches the best known ratio for unit weights of [34]. For unit disk graphs we improve the ratiothis is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln 2 k)/ when m ≥ (1 + )k; furthermore, we obtain ratio min m m−k , √ k • O(ln 2 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. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms Keywords and phrases k-connected graph, m-dominating set, approximation algorithm, rooted subset k-connectivity, subset k-connectivity Digital Object Identifier 10.4230/LIPIcs.ESA.2020.73 Acknowledgements I thank an anonymous referee for many useful comments.The problem generalizes several classic problems including Set-Cover (k = 0, m = 1), Set-Multicover (k = 0), and Connected Dominating Set (k = m = 1). The Connected Dominating Set problem is closely related to the Node Weighted Steiner Tree problem, and both problems admit a tight ratio O(log n) [16,12,13]. In unit disk graphs, the problem is NP-hard [5], admits a PTAS for unit weights [3], and ratio 3+2.5ρ+ for arbitrary weights [33,35], where ρ is the ratio for the edge-weighted Steiner Tree problem in general graphs. The (k, m)-CDS problem models (fault tolerant) virtual backbones in networks [7,6], and it was studied extensively, c.f.

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