Approximating Steiner networks with node weights

• Authors: Zeev Nutov
• Venue: LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
• Year: 2008

The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and edge-connectivity requirements {r(u, v) : u, v ∈ U ⊆ 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 [12], 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 ċ O(ln |U|), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [14] for the case rmax = 1. We also give an O(ln |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. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for |U| = 2 and unit weights.