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Bipartite perfect matching as a real polynomial

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Abstract

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full total degree and \((1 - o(1)) \cdot {2^{{n^2}}}\) monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(n log n). Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.

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Acknowledgements

We thank Nati Linial for helpful discussions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 740282).

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Authors and Affiliations

  1. School of Computer Science and Engineering, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Gal Beniamini & Noam Nisan

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  1. Gal Beniamini
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  2. Noam Nisan
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Correspondence to Gal Beniamini.

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Beniamini, G., Nisan, N. Bipartite perfect matching as a real polynomial. Isr. J. Math. 256, 91–131 (2023). https://doi.org/10.1007/s11856-023-2505-9

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  • DOI: https://doi.org/10.1007/s11856-023-2505-9

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