Approximating subset k-connectivity problems

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A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T; T is k-connected in J if T is k-connected to every s ∈ T. We consider the Subset k-Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, a node subset T ⊆ V, and a subgraph J = (V,E J ) of G such that T is (k-1)-connected in J, find a minimum-cost augmenting edge-set F ⊆ E\E J such that T is k-connected in J∪F. The problem admits trivial ratio O(|T| 2). We consider the case |T| > k and prove that for directed/undirected graphs and edge/node-costs, a ρ-approximation algorithm for Rooted Subset k -Connectivity Augmentation implies the following approximation ratios for Subset k -Connectivity Augmentation: (i) b(ρ + k) + (|T|/|T|-k) 2O (log |T|/|T|-k) and (ii) ρ·O (|T|/|T|-k log k), where b = 1 for undirected graphs and b = 2 for directed graphs. The best known values of ρ on undirected graphs are min {|T|,O(k)} for edge-costs and min {|T|,O(k log|T|)} for node-costs; for directed graphs ρ = |T| for both versions. Our results imply that unless k = |T| - o(|T|), Subset k-Connectivity Augmentation admits the same ratios as the best known ones for the rooted version. This improves the ratios in [19,14].

Original languageEnglish
Title of host publicationApproximation and Online Algorithms - 9th International Workshop, WAOA 2011, Revised Selected Papers
Number of pages12
StatePublished - 2012
Event9th International Workshop on Approximation and Online Algorithms, WAOA 2011 - Saarbrucken, Germany
Duration: 8 Sep 20119 Sep 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7164 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference9th International Workshop on Approximation and Online Algorithms, WAOA 2011


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