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Extended formulations from communication protocols in output-efficient time

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Abstract

Deterministic protocols are well-known tools to obtain extended formulations, with many applications to polytopes arising in combinatorial optimization. Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the number of rows of the slack matrix (hence, on the exact description in the original space). We give general sufficient conditions under which those tools can be implemented as to be output-efficient, showing applications to e.g. Yannakakis’ extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known. For specific classes of polytopes, we give also a direct, efficient construction of extended formulations arising from protocols. Finally, we deal with extended formulations coming from unambiguous non-deterministic protocols.

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Acknowledgements

We thank Mihalis Yannakakis for inspiring discussions and Samuel Fiorini for useful comments on [1], where preliminary versions of certain results presented here had appeared. Manuel Aprile would also like to thank Aurélie Lagoutte and Nicolas Bousquet for useful discussions. We are also indebted to the anonymous referees for their suggestions and corrections, which helped us to improve the structure and the presentation of the paper, and in particular to one of them for pointing us to the reference [7].

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  1. Université Libre de Bruxelles, Bruxelles, Belgium

    Manuel Aprile

  2. IEOR, Columbia University, New York, USA

    Yuri Faenza

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  1. Manuel Aprile
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  2. Yuri Faenza
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Correspondence to Manuel Aprile.

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Aprile, M., Faenza, Y. Extended formulations from communication protocols in output-efficient time. Math. Program. 183, 41–59 (2020). https://doi.org/10.1007/s10107-020-01535-9

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