Lagrangean decomposition/relaxation for the routing and wavelength assignment problem

This work deals with solving the Routing and Wavelength Assignment problem where the number of accepted connections is to be maximized. Lagrangean decomposition as well as Lagrangean relaxation are studied for both node‐arc formulations and arc‐path formulation. It is shown that relaxing the demand constraints yields an edge‐disjoint path problem with profits that is NP‐hard, while the Lagrangean problem obtained when the clash constraints are relaxed becomes a shortest path problem or a 0–1 linear knapsack problem, depending on the formulation used. Moreover, it is shown that the Lagrangean decomposition is not stronger than the Lagrangean relaxation of the demand constraints. We also propose a subgradient algorithm to solve the Lagrangean dual obtained by relaxing the clash constraints. Numerical results demonstrate a high quality of Lagrangean dual bounds with fast computation time.