Wavelength Assignment in Optical Networks with Fixed Fiber Capacity

We consider the problem of assigning wavelengths to demands in an optical network of m links. We assume that the route of each demand is fixed and the number of wavelengths available on a fiber is some parameter μ. Our aim is to minimize the maximum ratio between the number of fibers deployed on a link e and the number of fibers required on the same link e when wavelength assignment is allowed to be fractional. Our main results are negative ones. We show that there is no constantfactor approximation algorithm unless NP⊆ZPP. No such negative result is known if the routes are not fixed. In addition, unless all languages in NP have randomized algorithms with expected running time O(n), we show that there is no log μ approximation for any γ ∈ (0, 1) and no log m approximation for any γ ∈ (0, 0.5). Our analysis is based on hardness results for the problem of approximating the chromatic number in a graph. On the positive side, we present algorithms with approximation ratios O(logm + log μ), O(logDmax + log μ) and O(Dmax) respectively. Here Dmax is the length of the longest path. We conclude by presenting two variants of the problem and discussing which of our results still apply.