Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Polyhedral techniques in combinatorial optimization

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

The Windy General Routing Polyhedron: A global view of many known Arc Routing Polyhedra

[EN] The windy postman problem consists of finding a minimum cost traversal of all of the edges of an undirected graph with two costs associated with each edge, representing the costs of traversing it in each direction. In this paper we deal with the windy general routing problem (WGRP), in which only a subset of edges must be traversed and a subset of vertices must be visited. This is also an NP-hard problem that generalizes many important arc routing problems (ARPs) and has some interesting real-life applications. Here we study the description of the WGRP polyhedron, for which some general properties and some large families of facet-inducing inequalities are presented. Moreover, since the WGRP contains many well-known routing problems as special cases, this paper also provides a global view of their associated polyhedra. Finally, for the first time, some polyhedral results for several ARPs defined on mixed graphs formulated by using two variables per edge are presented.This work was supported by the Ministerio de Educación y Ciencia of Spain (project MTM2006-14961-C05-02).Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2008). The Windy General Routing Polyhedron: A global view of many known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics. 22(2):606-628. https://doi.org/10.1137/050640886S60662822