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The Approximate Degree of Bipartite Perfect Matching

Author Gal Beniamini

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Gal Beniamini
  • The Hebrew University of Jerusalem, Israel

Acknowledgements

I would like to thank Noam Nisan and Nati Linial for helpful discussions. I would also like to thank Bruno Loff for pointing out typographical errors in an earlier version.

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Gal Beniamini. The Approximate Degree of Bipartite Perfect Matching. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.1

Abstract

The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the 𝓁_∞-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching of order n, is Θ̃(n^(3/2)). The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small 𝓁₁-norm - only exponential in Θ(n log n). The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube [Aaronson et al., 2021; Huang, 2019]. We show that the spectral sensitivity of perfect matching is exactly Θ(n^(3/2)).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Oracles and decision trees
Keywords
  • Bipartite Perfect Matching
  • Boolean Functions
  • Approximate Degree

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