Consider a requirement graph whose vertices represent customers and an edge represents the need to route a unit of flow between its end vertices along a single path. All these flows are to be routed simultaneously. A solution network consists of a (multi)graph on the same set of vertices, such that it is possible to route simultaneously all of the required flows in such a way that no edge is used more than K times. The Synthesis of Wavelength Routing Network (SWRN) problem is to compute a solution network of a minimum number of edges. This problem has significant importance in the world of fiber-optic networks where a link can carry a limited amount of different wavelengths and one is interested in finding a minimum-cost network such that all the requirements can be carried in the network without changing the wavelength of a path at any of its internal vertices. In this paper, we prove that the SWRN problem is NP-hard for any constant K (K ≥ 2). Then, we assume that GR is a clique with n vertices and we find an ‘almost’ optimal solution network for all values of K (K = o(n)) and present a Min{(K + 1)/2, 2 + 2/(K − 1)}-approximation algorithm for the general case and a 2-approximation algorithm for d-regular graphs.
