A relax and cut approach using the multi-commodity flow formulation for the traveling salesman problem

En este artículo nosotros exploramos una formulación de flujo multiproductos para el Problema del Agente Viajero Asimétrico (ATSP) en la obtención de cotas duales de este problema. El procedimiento empleado es una variante del método relax and cut propuesto en la literatura que computa los multiplicadores lagrangianos asociados a las restricciones de eliminación de subrutas preservando la optimalidad de los multiplicadores asociados a las restricciones de asignación. Los resultados obtenidos con la experimentación computacional son alentadores y muestran que el algoritmo propuesto genera buenas cotas duales con un tiempo de ejecución bajo. ABSTRACT In this paper we explore the multi-commodity flow formulation for the Asymmetric Traveling Salesman Problem (ATSP) to obtain dual bounds. The procedure employed is a variant of a relax and cut procedure proposed in the literature that computes the Lagrangean multipliers associated to the subtour elimination constraints preserving the optimality of the multipliers associated to the assignment constraints. The results obtained by the computational study are encouraging and show that the proposed algorithm generated good dual bounds for the ATSP with a low execution time

Exact Algorithms for the Vertex Separator Problem in Graphs

In this article, we propose a Lagrangian relaxation framework to solve the vertex separator problem (VSP). This framework is based on the development of relax-and-cut algorithms which embed the separation of valid inequalities for the VSP discussed by Bales and de Souza (Math Program 103 (2005), 583-608) in the subgradient method. These relax-and-cut algorithms are then used as a preprocessing phase in a hybrid algorithm which combines them with branch-and-cut algorithms proposed by de Souza and Bales (Math Program 103 (2005), 609-631). This is done basically by feeding the branch-and-cut algorithms not only with the primal bound but also the cuts separated during the preprocessing phase. Computational results obtained with benchmarks from the literature showed that the hybrid algorithm developed here outperforms the best exact algorithm available for the VSP to date. (C) 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 57(3), 212-230 2011573SI21223