Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations

TL;DR

This paper establishes fundamental lower bounds on the size of mixed-integer linear formulations for key combinatorial polytopes, indicating that compact representations require many integer variables, impacting the design of optimization models.

Contribution

It proves a new lower bound on the number of integer variables needed in polynomial-sized mixed-integer formulations for the matching and traveling salesman polytopes.

Findings

Any polynomial-sized mixed-integer formulation for the matching polytope requires at least (rac{n}{\u221a{\u001f n}}) integer variables.

The lower bound extends to the traveling salesman polytope via known reductions.

Implications include that many classic routing and matching problems cannot have small mixed-integer formulations with few integer variables.

Abstract

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $\Omega(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first non-trivial lower bound on the number of integer variables needed in such settings.