Approximation Results for Wavelength Routing in Directed Trees

The paper deals with wavelength routing in WDM (wavelength division multiplexing) optical networks. Wavelengths must be assigned to connection requests which are represented as directed paths, and it is required that paths receive different wavelengths if they share a directed link. The goal is to minimize the number of wavelengths used. Mihail et al. [10] conjectured that there is a solution for directed trees with at most b 2Lmaxc wavelengths where Lmax is the maximum load on a directed link. In this paper, we show that there exists a binary tree with load Lmax = 3 such that every solution needs at least 5 wavelengths. Furthermore, we present an approximation algorithm for binary trees with approximation ratio b 3Lmaxc b 3OPT c. This improves the bound b 7 4Lmaxc for binary trees given by Mihail et al. [10]. Finally, we show that no local greedy algorithm for binary trees can go below b 3Lmaxc. This ratio is the best ratio that can be achieved by any local greedy algorithm for this problem.