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Algebraic Representations of Unique Bipartite Perfect Matching

Author Gal Beniamini



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Gal Beniamini
  • The Hebrew University of Jerusalem, Israel

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Gal Beniamini. Algebraic Representations of Unique Bipartite Perfect Matching. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.16

Abstract

We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results [Beniamini, 2020; Beniamini and Nisan, 2021], we show that, surprisingly, the dual description is sparse and has low 𝓁₁-norm - only exponential in Θ(n log n), and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula. These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight Θ(n log n) bound on the log-rank of the associated two-party communication task.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • Bipartite Perfect Matching
  • Boolean Functions
  • Partially Ordered Sets

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