Approximating k-Connected m-Dominating Sets

Abstract

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.

Improved Approximation Algorithm for Minimum-Weight $(1,m)$--Connected Dominating Set

“…However, as explained in the previous section, the analysis in [25] is flawed. By reducing the problem to a rooted subset k-connectivity instance, Nutov proposed an O(k 2 ln n)-approximation algorithm for the MinW(k, m)CDS problem in general graphs for k ≤ m [15], and then improved it to O(k ln n) in [16]. However, the constant in the big O of Nutov's algorithm is large, even for k = 1.…”

2-Node-Connectivity Network Design

“…Problems of finding low size/weight k-connected m-dominating set were vastly studied, but in general graphs non-trivial approximation algorithms are known only for the easier case m ≥ k, c.f. [21,18,37,35,5] and the references therein. For m < k, constant ratios are known for unit disk graphs and k ≤ 3 and m ≤ 2 [36], and ratio Õ(log 2 n) for 2-Edge-Connected Dominating Set in general graphs [5].…”