Uncapacitated Flow-based Extended Formulations

TL;DR

This paper investigates the complexity of extended formulations for various polytopes, proving exponential lower bounds for flow-based formulations and introducing new examples, advancing understanding in polyhedral combinatorics.

Contribution

It establishes exponential lower bounds on uncapacitated flow-based extended formulations for key polytopes and provides new flow-based formulations for 0/1-polytopes from regular languages.

Findings

Exponential lower bounds for perfect matching and TSP polytopes

New flow-based formulations for 0/1-polytopes from regular languages

Open problems in extended formulation complexity

Abstract

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems.