Bipartite perfect matching as a real polynomial

Gal Beniamini; Noam Nisan

doi:10.1007/s11856-023-2505-9

Bipartite perfect matching as a real polynomial

Gal Beniamini^*, Noam Nisan

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

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))⋅2n2 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.

Original language	American English
Pages (from-to)	91-131
Number of pages	41
Journal	Israel Journal of Mathematics
Volume	256
Issue number	1
DOIs	https://doi.org/10.1007/s11856-023-2505-9
State	Published - Sep 2023

Bibliographical note

Funding Information:
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).

Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.

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N2 - 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))⋅2n2 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.

AB - 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))⋅2n2 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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Bipartite perfect matching as a real polynomial

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Bibliographical note

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