Split cuts from sparse disjunctions

Split cuts are arguably the most effective class of cutting planes within a branch-and-cut framework for solving general Mixed-Integer Programs (MIP). Sparsity, on the other hand, is a common characteristic of MIP problems, and it is an important part of why the simplex method works so well inside branch-and-cut. In this work, we evaluate the strength of split cuts that exploit sparsity. In particular, we show that restricting ourselves to sparse disjunctions—and furthermore, ones that have small disjunctive coefficients—still leads to a significant portion of the total gap closed with arbitrary split cuts. We also show how to exploit sparsity structure that is implicit in the MIP formulation to produce splits that are sparse yet still effective. Our results indicate that one possibility to produce good split cuts is to try and exploit such structure.