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Uncapacitated Flow-based Extended Formulations
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
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