TY - GEN

T1 - Approximating minimum cost connectivity problems via uncrossable bifamilies and spider-cover decompositions

AU - Nutov, Zeev

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2009

Y1 - 2009

N2 - We give approximation algorithms for the Generalized Steiner Network (GSN) problem. The input consists of a graph G = (V, E) with edge/node costs, a node subset S ⊆ V, and connectivity requirements {r(s,t): s,t ∈ T ⊆ V}. The goal is to find a minimum cost subgraph H that for all s, t ∈ T contains r(s,t) pairwise edge-disjoint st-paths so that no two of them have a node in S - {s,t} in common. Three extensively studied particular cases are: Edge-GSN (S = ø), Node-GSN (S = V), and Element-GSN (r(s,t) = 0 whenever s ∈ S or t ∈ S). Let k = maxS,t∈T r(s,t). In Rooted GSN there is s ∈ T so that r(u, t) = 0 for all u ≠ s, and in the Subset k-Connected Subgraph problem r(s, t) = k for all s,t ∈ T. For edge costs, our ratios are O(k2) for Rooted GSN and O(k2 logk) for Subset k-Connected Subgraph. This improves the previous ratio O(k2 log n) and settles the approximability of these problems to a constant for bounded k. For node-cost, our ratios are: • O(k log|T|) for Element-GSN, matching the best known ratio for Edge-GSN. • O(k2 log |T|) for Rooted GSN and O(k3 log |T|) for Subset k-Connected Subgraph, improving the ratio O(k8 log2 |T|). • O(k 4log2 |T|) for GSN; this is the first non-trivial approximation algorithm for the problem.

AB - We give approximation algorithms for the Generalized Steiner Network (GSN) problem. The input consists of a graph G = (V, E) with edge/node costs, a node subset S ⊆ V, and connectivity requirements {r(s,t): s,t ∈ T ⊆ V}. The goal is to find a minimum cost subgraph H that for all s, t ∈ T contains r(s,t) pairwise edge-disjoint st-paths so that no two of them have a node in S - {s,t} in common. Three extensively studied particular cases are: Edge-GSN (S = ø), Node-GSN (S = V), and Element-GSN (r(s,t) = 0 whenever s ∈ S or t ∈ S). Let k = maxS,t∈T r(s,t). In Rooted GSN there is s ∈ T so that r(u, t) = 0 for all u ≠ s, and in the Subset k-Connected Subgraph problem r(s, t) = k for all s,t ∈ T. For edge costs, our ratios are O(k2) for Rooted GSN and O(k2 logk) for Subset k-Connected Subgraph. This improves the previous ratio O(k2 log n) and settles the approximability of these problems to a constant for bounded k. For node-cost, our ratios are: • O(k log|T|) for Element-GSN, matching the best known ratio for Edge-GSN. • O(k2 log |T|) for Rooted GSN and O(k3 log |T|) for Subset k-Connected Subgraph, improving the ratio O(k8 log2 |T|). • O(k 4log2 |T|) for GSN; this is the first non-trivial approximation algorithm for the problem.

KW - Approximation algorithms

KW - Generalized steiner network

UR - http://www.scopus.com/inward/record.url?scp=77952393163&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2009.9

DO - 10.1109/FOCS.2009.9

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AN - SCOPUS:77952393163

SN - 9780769538501

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 417

EP - 426

BT - Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

T2 - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

Y2 - 25 October 2009 through 27 October 2009

ER -