Inapproximability Results for Wavelength Assignment in WDM Optical Networks

We address the issue of inapproximability of the wavelength assignment problem in wavelength division multiplexing (WDM) optical networks. We prove that in an n-node WDM optical network with m lightpaths and maximum load L, if NP = ZPP, for any constant δ > 0, no polynomial time algorithm can achieve approximation ratio n1/2−δ or m1−δ , where NP is the class of problems which can be solved by nondeterministic polynomial time algorithms, and ZPP is the class of problems that can be solved by polynomial randomized algorithms with zero probability of error. Furthermore, the above result still holds even when L = 2. We also prove that no algorithm can guarantee the number of wavelengths to be less than ( √ n/2)L or (m/2)L. This is the first time inapproximability results are established for the wavelength assignment problem in WDM optical networks. We also notice the following fact, namely, there is a polynomial time algorithm for wavelength assignment which achieves approximation ratio of O(m(log logm)/(logm)). Therefore, the above lower bound of m1−δ is nearly tight.