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Theory of Computing Systems

, Volume 62, Issue 3, pp 510–532 | Cite as

Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

  • Zeev Nutov
Article

Abstract

Let κ G (s, t) denote the maximum number of pairwise internally disjoint st-paths in a graph G = (V, E). For a set \(T \subseteq V\) of terminals, G is k-T-connected if κ G (s, t) ≥ k for all s, tT; if T = V then G is k-connected. Given a root node s, G is k- (T, s)-connected if κ G (t, s) ≥ k for all tT. We consider the corresponding min-cost connectivity augmentation problems, where we are given a graph G = (V, E) of connectivity k, and an additional edge set \(\hat E\) on V with costs. The goal is to compute a minimum cost edge set \(J \subseteq \hat {E}\) such that \(G \cup J\) has connectivity k + 1. For the k-T-Connectivity Augmentation problem when \(\hat {E}\) is an edge set on T we obtain ratio \(O\left (\ln \frac {|T|}{|T|-k}\right )\), improving the ratio \(O\left (\frac {|T|}{|T|-k} \cdot \ln \frac {|T|}{|T|-k}\right )\) of Nutov (Combinatorica, 34(1), 95–114, 2014). For the k -Connectivity Augmentation problem we obtain the following approximation ratios. For n ≥ 3k − 5, we obtain ratio 3 for directed graphs and 4 for undirected graphs, improving the previous ratio 5 of Nutov (Combinatorica, 34(1), 95–114, 2014). For directed graphs and k = 1, or k = 2 and n odd, we further improve to 2.5 the previous ratios 3 and 4, respectively. For the undirected 2-(T, s)-Connectivity Augmentation problem we achieve ratio \(4\frac {2}{3}\), improving the previous best ratio 12 of Nutov (ACM Trans. Algorithms, 9(1), 1, 2014). For the special case when all the edges in \(\hat E\) are incident to s, we give a polynomial time algorithm, improving the ratio \(4\frac {17}{30}\) of Kortsarz and Nutov, (2015) and Nutov (Algorithmica, 63(1-2), 398–410, 2012) for this variant.

Keywords

Node-connectivity augmentation Approximation algorithm Crossing biset family 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.The Open University of IsraelRa’ananaIsrael

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