Zero-one IP problems: Polyhedral descriptions & cutting plane procedures

A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie

A polyhedral approach for the generalized assignment problem.

The generalized assignment problem (GAP) consists of finding a maximal profit assignment of n jobs over m capacity constrained agents, whereby each job has to be processed by only one agent. This contribution approaches the GAP from the polyhedral point of view. A good upper bound is obtained by approximating the convex hull of the knapsack constraints in the GAP-polytope using theoretical work of Balas. Based on this result, we propose a procedure for finding close-to-optimal solutions, which gives us a lower bound. Computational results on a set of 60representative and highly capacitated problems indicate that these solutions lie within 0.06% of the optimum. After applying some preprocessing techniques and using the obtained bounds, we solve the generated instances to optimality by branch and bound within reasonable computing time.Assignment;

The complexity of lifted inequalities for the knapsack problem

AbstractIt is well known that one can obtain facets and valid inequalities for the knapsack polytope by lifting simple inequalities associated with minimal covers. We study the complexity of lifting. We show that recognizing integral lifted facets or valid inequalities can be done in O(n2) time, even if the minimal cover from which they are lifted is not given. We show that the complexities of recognizing nonintegral lifted facets and valid inequalities are similar, respectively, to those of recognizing general (not necessarily lifted) facets and valid inequalities. Finally, we show that recognizing valid inequalities is in co- NPC while recognizing facets is in Dn. The question of whether recognizing facets is complete for Dn is open