Polyhedral structure and efficient algorithms for certain classes of the directed rural postman problem

In this thesis we study a class of combinatorial optimization problems known as the Directed Rural Postman Problem (DRPP). In DRPP the objective is to find the minimum cost tour that traverses a predetermined set of arcs in a directed graph. We first provide three different integer programming formulations for the DRPP. Then, we study its polyhedral structure. In particular, we use Linear Programming techniques, Strong Cutting Planes methods as well as Branch and Bound procedures to obtain exact solutions to several classes of the DRPP that arise in many applications. Furthermore, for these classes of the DRPP, we characterize the constraints that induce facets and solve the corresponding separation problem. Finally, by exploring the sparse nature of the underlying directed graphs we study four special cases of the DRPP and show that the Linear Programming relaxation of the integer programming formulation is integral by using duality theory. We also prove that the balance and connect heuristic for the Rural Postman Problem gives the optimal solution for these special cases.