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

Abstract

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.

Approximating k-Connected m-Dominating Sets

“…Our main results is: For general graphs our ratio O(k ln n) improves the previous ratio O(k 2 ln n) of [26] and matches (while using totally different techniques) the best known ratio for unit weights of Zhang et. al.…”

“…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 ratio…”

“…[34]. For unit disk graphs our ratio min k m−k , k 2/3 • O(ln 2 k) improves the previous best ratio O(k ln k) of [26]; this is the first sublinear ratio for the problem, and for any constant > 0 and m = k(1 + ) the first polylogarithmic ratio O(ln 2 k)/ .…”

“…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.…”

“…survey we refer the reader to recent papers of Zhang, Zhou, Mo, and Du [31] and of Fukunaga [10], where they obtained in unit disk graphs ratios O(k 3 ln k) and O(k 2 ln k), respectively. This was improved to O(k ln k) in [26], where is also given ratio O(k 2 ln n) in general graphs.…”