Wavelength Management in WDM Rings to Maximize the Number of Connections
Ioannis Caragiannis · SIAM Journal on Discrete Mathematics · 2009
We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a wavelength division ultiplexing (WDM) optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of w wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set of connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring (maxRPC) problem. Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of the solution computed by such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain upper bounds on the approximation ratio of several algorithms. The best bounds obtained are $4/3$ and $103/73\approx1.41096$ for undirected and bidirected rings; these results improve known bounds of $3/2$ and $11/7\approx1.57143$, respectively. Our results extend to multifiber ring networks and also apply to the maximum path coloring (maxPC) problem, where paths instead of connection requests are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit. We present an algorithm based on linear programming and randomized rounding with approximation ratio 1.49015, improving on the $\frac{e}{e-1}\approx1.58198$ bound obtained by a simple iterative algorithm.