Polyhedral Approaches
Résumé
This chapter discusses polyhedral approaches in combinatorial optimization. Using a cutting-plane algorithm, it is possible that an optimal solution of the problem will not be obtained. The chapter introduces the basic elements of polyhedral theory. It discusses the relationship between combinatorial optimization and linear programming. The chapter focuses on some proof techniques for polyhedra, in particular it introduce methods for proving that a given inequality defines a facet of a combinatorial polyhedron, and that a linear system describes an integer polyhedron. The chapter discusses integer polyhedra and min–max relationships in combinatorial optimization. In particular, totally unimodular matrices, totally dual integral systems, and blocking and antiblocking polyhedra. The chapter examines cut, and branch-and-cut, methods and the relationship between separation and optimization. It presents some applications of these techniques to the maximum cut and survivable network design problems.