Cutting Planes for Binarized Integer Programs

TL;DR

The paper investigates how different binarization strategies for bounded general-integer variables in network-flow IPs influence MILP solver performance. By comparing full, unary, and logarithmic binarizations and introducing MIR cuts derived from flow-conservation constraints, the study reveals substantial performance gains from certain formulations (notably full binarization with strengthening) and highlights the pivotal role of how constraints are represented. The authors extend the analysis to CMST and demonstrate that MIR-based formulation cuts can dramatically reduce LP gaps, often outperforming default solver cuts across CPLEX and Gurobi. These findings provide practical guidelines for choosing binarizations and crafting cuts to improve MIP solver efficiency in network-flow contexts.

Abstract

We consider integer programming problems with bounded general-integer variables belonging to the general class of network flow problems. For those, we computationally investigate the effect on mixed-integer linear programming (MIP) solvers of the different ways of producing extended formulations that replace a bounded general integer variable by a linear combination of a set of auxiliary binary variables linked by additional linear constraints. We show that MILP solvers perform very differently depending on which extended formulations is used and we interpret that different performance through the lens of cutting planes generation. Finally, we discuss a simple family of mixed-integer rounding inequalities that especially benefit from the reformulation, and we show its benefit within different MIP solvers. This provides methodological and practical guidelines for the use of those extended formulations in MIP and, to the best of our knowledge, this is the first extensive computational analysis of the topic.