Bipartite perfect matching as a real polynomial

Gal Beniamini; Noam Nisan

doi:10.1145/3406325.3451002

Bipartite perfect matching as a real polynomial

Gal Beniamini, Noam Nisan

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

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 degree and (1-on(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 I(n logn). Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered"is Eulerian.

Original language	American English
Title of host publication	STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Editors	Samir Khuller, Virginia Vassilevska Williams
Publisher	Association for Computing Machinery
Pages	1118-1131
Number of pages	14
ISBN (Electronic)	9781450380539
DOIs	https://doi.org/10.1145/3406325.3451002
State	Published - 15 Jun 2021
Event	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021

Publication series

Name	STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING
ISSN (Print)	0737-8017

Conference

Conference	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/Territory	Italy
City	Virtual, Online
Period	21/06/21 → 25/06/21

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:
© 2021 ACM.

Keywords

Bipartite Perfect Matching
Boolean Functions
Elementary Graphs

Access to Document

10.1145/3406325.3451002

Cite this

Beniamini, G., & Nisan, N. (2021). Bipartite perfect matching as a real polynomial. In S. Khuller, & V. V. Williams (Eds.), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (pp. 1118-1131). (STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING). Association for Computing Machinery. https://doi.org/10.1145/3406325.3451002

Beniamini, Gal ; Nisan, Noam. / Bipartite perfect matching as a real polynomial. STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. editor / Samir Khuller ; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. pp. 1118-1131 (STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING).

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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 degree and (1-on(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 I(n logn). Our proof relies heavily on the fact that the lattice of graphs which are {"}matching-covered{"}is Eulerian.",

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language = "American English",

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Beniamini, G & Nisan, N 2021, Bipartite perfect matching as a real polynomial. in S Khuller & VV Williams (eds), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING, Association for Computing Machinery, pp. 1118-1131, 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, Virtual, Online, Italy, 21/06/21. https://doi.org/10.1145/3406325.3451002

Bipartite perfect matching as a real polynomial. / Beniamini, Gal; Nisan, Noam.
STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. ed. / Samir Khuller; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. p. 1118-1131 (STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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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 degree and (1-on(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 I(n logn). 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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Keywords

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