On the knapsack closure of 0-1 Integer Linear Programs

Abstract

Abstract Many inequalities for Mixed-Integer Linear Programs (MILPs) or pure Integer Linear Programs (ILPs) are derived from the Gomory corner relaxation, where all the nonbinding constraints at an optimal LP vertex are relaxed. Computational results show that the corner relaxation gives a good approximation of the integer hull for problems with general-integer variables, but the approximation is less satisfactory for problems with 0-1 variables only. A possible explanation is that, for 0-1 ILPs, even the non-binding variable bound constraints x j ⩾ 0 or x j ⩽ 1 play an important role, hence their relaxation produces weaker bounds. In this note we address a relaxation for 0-1 ILPs that explicitly takes all variable bound constraints into account. More specifically, we introduce the concept of knapsack closure as a tightening of the classical Chvatal-Gomory (CG) closure. The knapsack closure is obtained as follows: for all inequalities w T x ⩽ w 0 valid for the LP relaxation, add to the original system all the valid inequalities for the knapsack polytope c o n v { x ∈ { 0 , 1 } n : w T x ⩽ w 0 } . A MILP model for the corresponding separation problem is also introduced.