Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets: Theory and Computation

Abstract

We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have a number of interesting properties. First, they generalize the well-known (1, k)configuration inequalities. Second, they are not aggregation cuts. Third, they cannot be generated as a rank-1 Chvátal-Gomory cut from the inequality system consisting of the knapsack constraints and all their minimal cover inequalities. We also provide conditions under which the inequalities are facets for the convex hull of the totally-ordered knapsack set, as well as conditions for those inequalities to fully characterize its convex hull. We give an integer program to solve the separation and provide numerical experiments that showcase the strength of these new inequalities.

A very fast method for guaranteed generation of one facet for 0-1 knapsack polyhedron

“…Del Pia et al [21] proposed a new approach to generate valid inequalities for a special multiple knapsack set, called the totally-ordered multiple knapsack set (TOMKS). Bazzi et al [22] address the issue of relaxation of exponential size and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities.…”