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Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials

Authors Cornelius Brand, Kevin Pratt

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Parameterized Algorithms
  • Algebraic Algorithms
  • Longest Cycle
  • Matroid Parity

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Abstract

We study the following problem and its applications: given a homogeneous degree-d polynomial g as an arithmetic circuit C, and a d × d matrix X whose entries are homogeneous linear polynomials, compute g(∂/∂ x₁, …, ∂/∂ x_n) det X. We show that this quantity can be computed using 2^{ω d}|C|poly(n,d) arithmetic operations, where ω is the exponent of matrix multiplication. In the case that C is skew, we improve this to 4^d|C| poly(n,d) operations, and if furthermore X is a Hankel matrix, to φ^{2d}|C| poly(n,d) operations, where φ = (1+√5)/2 is the golden ratio.
Using these observations we give faster parameterized algorithms for the matroid k-parity and k-matroid intersection problems for linear matroids, and faster deterministic algorithms for several problems, including the first deterministic polynomial time algorithm for testing if a linear space of matrices of logarithmic dimension contains an invertible matrix. We also match the runtime of the fastest deterministic algorithm for detecting subgraphs of bounded pathwidth with a new and simple approach. Our approach generalizes several previous methods in parameterized algorithms and can be seen as a relaxation of Waring rank based methods [Pratt, FOCS19].

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Cornelius Brand and Kevin Pratt. Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.38

Author Details

Cornelius Brand
  • Charles University, Prague, Czech Republic
Kevin Pratt
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Brand, Cornelius: The research was supported by OP RDE project No. CZ.02.2.69/0.0/0.0/ 18_053/0016976 International mobility of research, technical and administrative staff at Charles University.

Acknowledgements

We would like to thank Ryan O'Donnell and several anonymous reviewers for their many helpful comments on earlier drafts of this paper. In particular we thank an anonymous reviewer for suggesting the name "totally multilinear."

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