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DOI: 10.4230/LIPIcs.MFCS.2022.16

Algebraic Representations of Unique Bipartite Perfect Matching

Authors: Gal Beniamini

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

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.

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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)

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@InProceedings{beniamini:LIPIcs.MFCS.2022.16,
  author =	{Beniamini, Gal},
  title =	{{Algebraic Representations of Unique Bipartite Perfect Matching}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.16},
  URN =		{urn:nbn:de:0030-drops-168140},
  doi =		{10.4230/LIPIcs.MFCS.2022.16},
  annote =	{Keywords: Bipartite Perfect Matching, Boolean Functions, Partially Ordered Sets}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.1

The Approximate Degree of Bipartite Perfect Matching

Authors: Gal Beniamini

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

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)).

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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)

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@InProceedings{beniamini:LIPIcs.CCC.2022.1,
  author =	{Beniamini, Gal},
  title =	{{The Approximate Degree of Bipartite Perfect Matching}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{1:1--1:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.1},
  URN =		{urn:nbn:de:0030-drops-165634},
  doi =		{10.4230/LIPIcs.CCC.2022.1},
  annote =	{Keywords: Bipartite Perfect Matching, Boolean Functions, Approximate Degree}
}

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Algebraic Representations of Unique Bipartite Perfect Matching

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The Approximate Degree of Bipartite Perfect Matching

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