Binary Extended Formulations

TL;DR

This paper investigates binary extended formulations for mixed-integer problems, demonstrating that unimodular formulations are the strongest, comparing various formulations, and showing that branching on binary variables can reduce enumeration tree size.

Contribution

It introduces the concept of unimodular extended formulations, compares their strength with other formulations, and analyzes their impact on branch-and-bound efficiency.

Findings

Unimodular extended formulations are the strongest among binary formulations.

Branching on binary variables can significantly reduce enumeration tree size.

Comparison of literature formulations highlights their relative strengths.

Abstract

We analyze different ways of constructing binary extended formulations of mixed-integer problems with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables, what we call "unimodular" extended formulations are the strongest. We also compare the strength of some binary extended formulations from the literature. Finally, we study the behavior of branch-and-bound on such extended formulations and show that branching on the new binary variables leads to significantly smaller enumeration trees in some cases.