Hardness and approximation of traffic grooming

Abstract

Traffic grooming is a central problem in optical networks. It refers to packing low rate signals into higher speed streams, in order to improve bandwidth utilization and reduce network cost. In WDM networks, the most accepted criterion is to minimize the number of electronic terminations, namely the number of SONET Add–Drop Multiplexers (ADMs). In this article we focus on ring and path topologies. On the one hand, we provide an inapproximability result for Traffic Grooming for fixed values of the grooming factor g , answering affirmatively the conjecture of Chow and Lin [T. Chow, P. Lin, The ring grooming problem, Networks 44 (2004), 194–202]. More precisely, we prove that Ring Traffic Grooming for fixed g ≥ 1 and Path Traffic Grooming for fixed g ≥ 2 are Apx-complete. That is, they do not accept a PTAS unless P = NP . Both results rely on the fact that finding the maximum number of edge-disjoint triangles in a tripartite graph (and more generally cycles of length 2 g + 1 in a ( 2 g + 1 ) -partite graph of girth 2 g + 1 ) is Apx-complete. On the other hand, we provide a polynomial-time approximation algorithm for Ring and Path Traffic Grooming, based on a greedy cover algorithm, with an approximation ratio independent of g . Namely, the approximation guarantee is O ( n 1 / 3 log 2 n ) for any g ≥ 1 , n being the size of the network. This is useful in practical applications, since in backbone networks the grooming factor is usually greater than the network size. Finally, we improve this approximation ratio under some extra assumptions about the request graph.