A polynomial time approximation scheme for the SONET ring loading problem

Abstract

A rapidly emerging design paradigm for large-scale optical fiber networks involves connecting clusters of nodes through a synchronous optical network (SONET) ring and building a network over these rings. Although the fiber itself offers virtually unlimited bandwidth, the add/drop multiplexers (ADMs) determine the actual bandwidth available along any edge of the SONET ring. Consequently, there is a cost involved in supporting a high bandwidth along the ring. An important optimization problem arises in this context: Given a set of nodes connected along a bidirectional SONET ring, we must determine a routing scheme that minimizes the bandwidth required to satisfy all the pairwise traffic demands. This problem, known as the ring loading problem, has been studied extensively in recent years. S. Cosares and I. Saniee reported in Telecommunications Systems (Volume 3, 1994) that this problem is NP-hard, which makes it unlikely that a polynomial time algorithm exists to compute an optimal solution. We therefore shift our attention to polynomial time approximation algorithms. Towards this end, Cosares and Saniee presented a polynomial time algorithm that approximates the optimal solution value to within a multiplicative factor of two. More recently, A. Schrijver, P. Seymour, and P. Winkler developed an efficient algorithm (documented in the SIAM Journal of Discrete Mathematics, 1997) that can compute a solution that exceeds the optimum by at most an additive term of 1.5d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> , where d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> is the largest traffic demand between any pair of nodes. While this additive term is relatively small, in many input instances it would represent a significant fraction of the optimal solution value and possibly a significant deviation from it. Is there a polynomial time approximation algorithm that allows us to achieve an error that is an arbitrarily small fraction of the optimal solution value? In this paper, we answer this question in the affirmative. More precisely, building on the work of Schrijver, Seymour, and Winkler, we show that for any fixed ɛ > 0, there exists a polynomial time algorithm that computes a solution that requires bandwidth at most (1 + ɛ) times the optimal bandwidth.