Improved Approximation Algorithms for Min-Cost Connectivity Augmentation Problems

Abstract

In Connectivity Augmentation problems we are given a graph H = (V, EH) and an edge set E on V , and seek a min-size edge set J ⊆ E such that H ∪ J has larger edge/node connectivity than H. In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by 1. In the Block-Tree Augmentation problem H is connected and H ∪ S should be 2-connected. In Leaf-to-Leaf Connectivity Augmentation problems every edge in E connects minimal deficient sets. For this version we give a simple combinatorial approximation algorithm with ratio 5/3, improving the 1.91 approximation of [3] (see also [16]), that applies for the general case. We also show by a simple proof that if the Steiner Tree problem admits approximation ratio α then the general version admits approximation ratio 1 + ln(4 − x) + , where x is the solution to the equation 1 + ln(4 − x) = α + (α − 1)x. For the currently best value of α = ln 4 + [4] this gives ratio 1.942. This is slightly worse than the ratio 1.91 of [3], but has the advantage of using Steiner Tree approximation as a "black box", giving ratio < 1.9 if ratio α ≤ 1.35 can be achieved.

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

“…3 [13], and β k = O(k ln k) for k ≥ 4 [11]. We also have β ′ k = O(k 2 ln n) by [11] and the correction of Vakilian [14] to the algorithm and the analysis of [11]; see also [6].…”

“…For various approximation algorithms for this problem see recent papers [5,4,9,6,3]. Specifically, Bansal [3] showed that if the problem of covering a γ-pliable familily achieves approximation ratio α, then (k, 3)-FGC admits approximation ratio 6 + α, thus obtaining a 22-approximation for (k, 3)-FGC.…”