Approximating k-Connected m-Dominating Sets

• Published in: Algorithmica Vol. 84; no. 6; pp. 1511 - 1525
• Main Author: Nutov, Zeev
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Approximation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Graphs; Mathematics of Computing; Minimum weight; Theory of Computation; Approximation algorithms; Subset; connectivity; connected graph; dominating set
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-00935-x

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 unit disk graphs we improve the ratio O ( k ln k ) of Nutov (Inf Process Lett 140:30–33, 2018) to min m 2 ( m - k + 1 ) 2 , 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 ) / ϵ 2 when m ≥ ( 1 + ϵ ) k ; furthermore, we obtain ratio min m m - k + 1 , k · 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.