TY - GEN

T1 - Approximating subset k-connectivity problems

AU - Nutov, Zeev

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=84859316007&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-29116-6_2

DO - 10.1007/978-3-642-29116-6_2

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:84859316007

SN - 9783642291159

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 9

EP - 20

BT - Approximation and Online Algorithms - 9th International Workshop, WAOA 2011, Revised Selected Papers

T2 - 9th International Workshop on Approximation and Online Algorithms, WAOA 2011

Y2 - 8 September 2011 through 9 September 2011

ER -