TY - JOUR

T1 - Approximating steiner networks with node-weights

AU - Nutov, Zeev

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

PY - 2010

Y1 - 2010

N2 - The (undirected) Steiner Network problem is as follows: given a graph G = (V, E) with edge/node-weights and edge-connectivity requirements {r(u,v): u,v ∈ U C V}, find a minimum-weight subgraph H of G containing U so that the uv-edge-connectivity in H is at least r(u, v) for all u,v ∈ U. The seminal paper of Jain [Combinatorial, 21 (2001), pp. 39-60], and numerous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum-weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(In |U|), where rmax = maxu,v ∈U r(u, v). This generalizes the result of Klein and Ravi [J. Algorithms, 19 (1995), pp. 104-115] for the case rmax = 1. We also give an O(In |U|)-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally disjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum node-weighted edge-cover of an uncrossable set-family. Finally, we give evidence that a polylogarithmic approximation ratio for NWSN with large rmax might not exist even for |U| = 2 and unit weights.

AB - The (undirected) Steiner Network problem is as follows: given a graph G = (V, E) with edge/node-weights and edge-connectivity requirements {r(u,v): u,v ∈ U C V}, find a minimum-weight subgraph H of G containing U so that the uv-edge-connectivity in H is at least r(u, v) for all u,v ∈ U. The seminal paper of Jain [Combinatorial, 21 (2001), pp. 39-60], and numerous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum-weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(In |U|), where rmax = maxu,v ∈U r(u, v). This generalizes the result of Klein and Ravi [J. Algorithms, 19 (1995), pp. 104-115] for the case rmax = 1. We also give an O(In |U|)-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally disjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum node-weighted edge-cover of an uncrossable set-family. Finally, we give evidence that a polylogarithmic approximation ratio for NWSN with large rmax might not exist even for |U| = 2 and unit weights.

KW - Approximation algorithms

KW - Intersecting families

KW - Node-weights

KW - Steiner networks

UR - http://www.scopus.com/inward/record.url?scp=77956019224&partnerID=8YFLogxK

U2 - 10.1137/080729645

DO - 10.1137/080729645

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AN - SCOPUS:77956019224

SN - 0097-5397

VL - 39

SP - 3001

EP - 3022

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 7

ER -