Cut-based Conflict Analysis in Mixed Integer Programming

TL;DR

This paper introduces a novel cut-based conflict analysis method for MIP solvers, leveraging linear combinations and cut generation, which improves solver performance over traditional graph-based methods.

Contribution

It proposes a new conflict analysis algorithm based on mixed integer rounding cuts, extending it to mixed binary and general MIP problems, outperforming existing methods.

Findings

Improved solver performance in terms of running time.

Reduced number of nodes in the search tree.

Higher instance solving rate.

Abstract

For almost two decades, mixed integer programming (MIP) solvers have used graph-based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this paper, we improve MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo-Boolean optimization. Phrased in MIP terminology, this type of conflict analysis can be understood as a sequence of linear combinations, integer roundings, and cut generation. We leverage this MIP perspective to design a new conflict analysis algorithm based on mixed integer rounding cuts, which theoretically dominates the state-of-the-art method in pseudo-Boolean optimization using Chv\'atal-Gomory cuts. Furthermore, we extend this cut-based conflict analysis from pure binary programs to mixed binary programs and-in limited form-to general MIP with also integer-valued variables. We perform an empirical evaluation of cut-based conflict analysis as implemented in the open-source MIP solver SCIP, testing it on a large and diverse set of MIP instances from MIPLIB 2017. Our experimental results indicate that the new algorithm improves the default performance of SCIP in terms of running time, number of nodes in the search tree, and the number of instances solved.