Approximating the traffic grooming problem

The problem of grooming is central in studies of optical networks, In graph-theoretic terms, this can be viewed as assigning colors to the lightpaths so that at most g of them (g being the grooming factor) can share one edge. The cost of a coloring is the number of optical switches (ADMs); each lightpath uses two ADM's, one at each endpoint, and in case g lightpaths of the same wavelength enter through the same edge to one node, they can all use the same ADM (thus saving g - 1 ADMs). The goal is to minimize the total number of ADMs. This problem was shown to be NP-complete for g = 1 and for a general g. Exact solutions a a re known for some specific cases, and approximation algorithms for certain topologies exist for g = 1. We present an approximation algorithm for this problem. For every value of g the running time of the algorithm is polynomial in the input size, and its approximation ratio for a wide variety of network topologies - including the ring topology - is shown to be 2 ln g + o(ln g). This is the first approximation algorithm for the grooming problem with a general grooming factor g.