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

TL;DR

This paper introduces a new class of multi-cover inequalities for totally-ordered multiple knapsack sets, providing theoretical properties, conditions for facet-defining inequalities, and computational evidence of their strength.

Contribution

It develops a novel method to generate and analyze multi-cover inequalities for totally-ordered knapsack sets, extending existing inequalities and offering new separation techniques.

Findings

The inequalities generalize (1, k)-configuration inequalities.

They are not aggregation cuts and cannot be generated as rank-1 Chvatal-Gomory cuts.

Numerical experiments show these inequalities significantly strengthen the formulation.

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 Chvatal-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.