TY - JOUR

T1 - Approximating k-Connected m-Dominating Sets

AU - Nutov, Zeev

N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - k-connected graph

KW - m-dominating set

KW - Subset k-connectivity

UR - http://www.scopus.com/inward/record.url?scp=85124831030&partnerID=8YFLogxK

U2 - 10.1007/s00453-022-00935-x

DO - 10.1007/s00453-022-00935-x

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AN - SCOPUS:85124831030

SN - 0178-4617

VL - 84

SP - 1511

EP - 1525

JO - Algorithmica

JF - Algorithmica

IS - 6

M1 - 6

ER -