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Uncapacitated flow-based extended formulations

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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.

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Notes

  1. We remark that although we allow for now \(Q\) to be unbounded, we will soon show that one can restrict to the case where \(Q\) is bounded, that is, a polytope.

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Acknowledgments

The authors thank Hans Tiwary for taking part in the early stage of this work, and Michele Conforti, Santanu Dey, Marco Di Summa, Sebastian Pokutta and Dirk Oliver Theis for stimulating discussions. Finally, we thank the two anonymous referees for carefully reading the manuscript and suggesting changes that contributed to improve the manuscript.

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Authors and Affiliations

  1. Département de Mathématique, Université libre de Bruxelles CP 216, Boulevard du Triomphe, 1050, Brussels, Belgium

    Samuel Fiorini

  2. Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121, Padova, Italy

    Kanstantsin Pashkovich

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  1. Samuel Fiorini
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  2. Kanstantsin Pashkovich
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Correspondence to Samuel Fiorini.

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Samuel Fiorini was partially supported by Fonds National de la Recherche Scientifique (F.R.S.-FNRS) and the Actions de Recherche Concertées (ARC) fund of the French community of Belgium. Kanstantsin Pashkovich was supported by the Progetto di Eccellenza 2008–2009 of the Fondazione Cassa Risparmio di Padova e Rovigo.

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Fiorini, S., Pashkovich, K. Uncapacitated flow-based extended formulations. Math. Program. 153, 117–131 (2015). https://doi.org/10.1007/s10107-015-0862-9

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