Abstract not found

We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.

Gomory's and Chvatal's cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is well-known that the Chvatal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvatal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the n-dimensional 0/1-cube is at most 3n² lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n³ lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coe#cients is O(n). The latter observation explains why for most cutting planes derived in polyhedral st...

Abstract We establish a precise correspondence between lift-and-project cuts for mixed 0-1 programs, simple disjunctive cuts (intersection cuts) and mixed-integer Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higher-dimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 0-1 polyhedron with respect to the above families of cutting planes.

Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, lift-and-project, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.

We extend the Sherali-Adams, Lovasz-Schrijver, Balas-Ceria-Cornuejols and Lasserre lift-and-project methods for 0-1 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a set-covering problem at least as strong as that given by the set of all valid inequalities with small coefficients, and, more generally, all valid inequalities where the right-hand side is not very large relative to the positive coefficients in the left-hand side. Applied to generalizations of vertex-packing problems, our methods yield, in polynomial time, relaxations that have unbounded rank using for example the N+ operator.

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure Integer Linear Program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal-Gomory (CG) separation problem as a Mixed Integer Linear Program (MILP) which is then solved by a general-purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory Mixed Integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal-Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, whose solution can typically be carried out in a reasonable amount of computing time.