Zero-Coefficient Cuts

Abstract

Many cuts used in practice to solve mixed integer programs are derived from a basis of the linear relaxation. Every such cut is of the form α ^T x ≥ 1, where x ≥ 0 is the vector of non-basic variables and α ≥ 0. For a point $\bar{x}$ of the linear relaxation, we call α ^T x ≥ 1 a zero-coefficient cut wrt. $\bar{x}$ if $\alpha^T \bar{x} = 0$, since this implies α _j  = 0 when $\bar{x}_j > 0$. We consider the following problem: Given a point $\bar{x}$ of the linear relaxation, find a basis, and a zero-coefficient cut wrt. $\bar{x}$ derived from this basis, or provide a certificate that shows no such cut exists. We show that this problem can be solved in polynomial time. We also test the performance of zero-coefficient cuts on a number of test problems. For several instances zero-coefficient cuts provide a substantial strengthening of the linear relaxation.