# An efficient approximation algorithm for the survivable network design problem

## Abstract

The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph of*n* vertices and connectivity requirements that are at most*k.* Both algorithms find a solution that is within a factor 2*k* − 1 of optimal for*k* ⩾ 2 and a factor 2 of optimal for*k* = 1. Our algorithm improves the time from O(*k*^{3}n^{4}) to O\((k^2 n^2 + kn^2 \sqrt {\log \log n} )\)). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the “redundant” edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (*s, t*) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

### Keywords

Network design Approximation algorithm Steiner tree## Preview

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