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During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.

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This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time. We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P|| w j T j problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P|| w j T j have been solved to optimality.

The Hop-Constrained Minimum Spanning Tree Problem (HMSTP) is a NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner Tree Problem (STP) in an adequate layered graph. We prove that the directed cut formulation for the STP defined in the layered graph, dominates (in terms of the linear programming relaxation) the best previously known formulations for the HMSTP. We then show how cuts in the extended layered graph space can be projected into new families of cuts in the original design space. We also adapt the proposed approach for the Diameter-Constrained Minimum Spanning Tree Problem (DMSTP). For situations when the diameter is odd we propose a new transformation into a single center problem that is quite effective. Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.

The Time Dependent Traveling Salesman Problem (TDTSP) is a generalization of the classical Traveling Salesman Problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.