On the complexity of the traffic grooming problem in optical networks

A central problem in optical networks is to assign wavelengths to a given set of lightpaths, so that at most g of them that share a physical link get the same wavelength (g is the grooming factor). The switching cost for each wavelength is the number of distinct endpoints of lightpaths of that wavelength, and the goal is to minimize the total switching cost. We prove NP-completeness results for the problem of minimizing the switching costs in path networks. First we prove that the problem is NP-complete in the strong sense, when all demands are either 0 or 1, the routing is single-hop, and the number of wavelengths is unbounded. Next we prove that the problem is NP-complete for any fixed g >= 2, and when the number of wavelengths is bounded. These results improve upon existing results regarding the complexity of the traffic grooming problem for ring and path networks.