Analysis of Sparse Cutting-planes for Sparse MILPs with Applications to Stochastic MILPs

TL;DR

This paper analyzes the effectiveness of sparse cutting-planes in improving the linear programming relaxations of sparse MILPs, providing bounds based on sparsity structure and exploring implications for stochastic MILPs.

Contribution

It introduces data-independent bounds on the strength of sparse cuts for various MILPs, including stochastic cases, based solely on sparsity structure.

Findings

Bounds depend only on sparsity structure and chosen supports.

Sparse cuts can significantly improve LP relaxations.

Results apply to stochastic MILPs with scenario-specific cuts.

Abstract

In this paper, we present an analysis of the strength of sparse cutting-planes for mixed integer linear programs (MILP) with sparse formulations. We examine three kinds of problems: packing problems, covering problems, and more general MILPs with the only assumption that the objective function is non-negative. Given a MILP instance of one of these three types, assume that we decide on the support of cutting-planes to be used and the strongest inequalities on these supports are added to the linear programming relaxation. Call the optimal objective function value of the linear programming relaxation together with these cuts as $z^{cut}$. We present bounds on the ratio of $z^{cut}$ and the optimal objective function value of the MILP that depends only on the sparsity structure of the constraint matrix and the support of sparse cuts selected, that is, these bounds are completely data independent. These results also shed light on the strength of scenario-specific cuts for two stage stochastic MILPs.