An efficient approximation algorithm for the survivable network design problem

The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-t ime approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435454) . 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 2k 1 of optimal for k ~> 2 and a factor 2 of optimal for k 1. Our algorithm improves the time from O(k3n 4) to O(k2n2 + kn21~dg log n). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435454) 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. * Corresponding author. E-mail: [email protected]. i A preliminary version of this paper has appeared in the Proceedings of the Third Mathematical Programming Society Conference on Integer Programming and Combinatorial Optimization, 1993, pp. 57-74. 2 E-mail: [email protected]. Research supported in part by NSF Grant No. CCR-9215199 and AT & T Bell Laboratories. 3 E-mail: [email protected]. Research supported in part by Air Force contracts AFOSR-89-0271 and F49620-92-J-0125 and DARPA contracts N00014-89-J-1988 and N00014-92-1799. 4 This research was performed while the author was a graduate student at MIT. Research supported by an NSF Graduate Fellowship, Air Force contract F49620-92-J-0125, DARPA contracts N00014-89-J1988 and N00014-92-J-1799, and AT & T Bell Laboratories. 0025-5610/98/$19.00 © 1998 The mathematical Programming Society, Inc. Published by Elsevier Science B.V. P H S 0 0 2 5 5 6 1 0 ( 9 7 ) 0 0 0 8 7 7 14 H.N. Gabow et al. / Mathematical Programming 82 (1998) 13-40