Wavelength Assignment in Multifiber WDM Star and Spider Networks

We consider the wavelength assignment problem in WDM optical networks with multiple parallel fibers: Given a set P of paths, assign a color to each path such that the number of paths with the same color on any link is at most the number of fibers in the link. Assuming the number of fibers in each link is fixed, we study two optimization problems. One is to minimize the number of colors for coloring P. The other is to color as many paths of P as possible with a given number of colors. The main results of the paper are: (1) Both the minimization and maximization problems are NP-hard in stars (thus in spiders) with uniform odd number of fibers. (2) The minimization problem is polynomial time solvable in stars with even number of fibers and in spiders with uniform even number of fibers. The result for spiders implies a (1 + 1/K - 1)-approximation algorithm for the minimization problem in spiders with uniform odd number k of fibers. (3) For the maximization problem, we show that it is polynomial time solvable for spiders with uniform even number of fibers and give a 1.58-approximation algorithm for spiders with arbitrary number of fibers. The algorithms for the maximization problem in spiders are based on our newly developed algorithm which optimally solves the call control problem in spiders. Call control is a well studied problem in communication networks. It is known solvable for stars but is NP-hard and MAX SNP-hard even for depth-3 trees. As the spider is a boundary topology between the star and the tree, the call control algorithm has its independent interests.