TY - JOUR

T1 - Approximating survivable networks with minimum number of steiner points

AU - Kamma, Lior

AU - Nutov, Zeev

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

PY - 2012/12

Y1 - 2012/12

N2 - Given a graph H = (U, E) and connectivity requirements r = {r(u,v): u, v ∈ R ⊆ U}, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, ∥·∥) and connectivity requirements, find a minimum size set S ⊂ M \ V of additional points, such that the unit disc graph induced by U = V S satisfies the requirements. In the (node-connectivity) Survivable Network Design Problem (SNDP) we are given a graph G = (V, E) with edge costs and connectivity requirements, and seek a minimum cost subgraph H of G that satisfies the requirements. Let k = max u,v∈V r(u, v) denote the maximum connectivity requirement. We will show a natural transformation of an SN-MSP instance (V, r) into an SNDP instance (G = (V, E), c, r), such that an α-approximation algorithm for the SNDP instance implies an α · O(k 2)-approximation algorithm for the SN-MSP instance. In particular, for the case of uniform requirements r(u, v) = k for all u, v ∈ V, we obtain for SN-MSP the ratio O(k 2 ln k), which solves an open problem from (Bredin et al. Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) (2005), 309-319).

AB - Given a graph H = (U, E) and connectivity requirements r = {r(u,v): u, v ∈ R ⊆ U}, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, ∥·∥) and connectivity requirements, find a minimum size set S ⊂ M \ V of additional points, such that the unit disc graph induced by U = V S satisfies the requirements. In the (node-connectivity) Survivable Network Design Problem (SNDP) we are given a graph G = (V, E) with edge costs and connectivity requirements, and seek a minimum cost subgraph H of G that satisfies the requirements. Let k = max u,v∈V r(u, v) denote the maximum connectivity requirement. We will show a natural transformation of an SN-MSP instance (V, r) into an SNDP instance (G = (V, E), c, r), such that an α-approximation algorithm for the SNDP instance implies an α · O(k 2)-approximation algorithm for the SN-MSP instance. In particular, for the case of uniform requirements r(u, v) = k for all u, v ∈ V, we obtain for SN-MSP the ratio O(k 2 ln k), which solves an open problem from (Bredin et al. Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) (2005), 309-319).

KW - approximation algorithms

KW - node-connectivity

KW - sensor networks

KW - unit disc graphs

UR - http://www.scopus.com/inward/record.url?scp=84868102776&partnerID=8YFLogxK

U2 - 10.1002/net.21466

DO - 10.1002/net.21466

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AN - SCOPUS:84868102776

SN - 0028-3045

VL - 60

SP - 245

EP - 252

JO - Networks

JF - Networks

IS - 4

ER -