Path Decomposition Under a New Cost Measure with Applications to Optical Network Design

We introduce a problem directly inspired by its application to DWDM (dense wavelength division multiplexing) network design. We are given a set of demands to be carried over a network. Our goal is to choose a route for each demand and to decompose the network into a collection of edge-disjoint simple paths. These paths are called optical line systems. The cost of routing one unit of demand is the number of line systems with which the demand route overlaps; our design objective is to minimize the total cost over all demands. This cost metric is motivated by the need to avoid O-E-O (optical-electrical-optical) conversions in optical transmission as well as to minimize the expense of the equipment necessary to carry the traffic.For given line systems, it is easy to find the optimal demand routes. On the other hand, for given demand routes designing the optimal line systems can be NP-hard. We first present a 2-approximation for general network topologies. As optical networks often have low node degrees, we also offer an algorithm that finds the optimal solution for the special case in which the node degree is at most 3.If neither demand routes nor line systems are fixed, the situation becomes much harder. Even for a restricted scenario on a 3-regular Hamiltonian network, no efficient algorithm can guarantee a constant approximation better than 2. For general topologies, we offer a simple algorithm with an O(log K)- and an O(log n)-approximation where K is the number of demands and n is the number of nodes. For rings, a common special topology, we offer a 3/2-approximation.