License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.38
URN: urn:nbn:de:0030-drops-141079
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14107/
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Brand, Cornelius ; Pratt, Kevin

Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials

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LIPIcs-ICALP-2021-38.pdf (0.7 MB)

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].

BibTeX - Entry

@InProceedings{brand_et_al:LIPIcs.ICALP.2021.38,
  author =	{Brand, Cornelius and Pratt, Kevin},
  title =	{{Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{38:1--38:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14107},
  URN =		{urn:nbn:de:0030-drops-141079},
  doi =		{10.4230/LIPIcs.ICALP.2021.38},
  annote =	{Keywords: Parameterized Algorithms, Algebraic Algorithms, Longest Cycle, Matroid Parity}
}

Keywords: Parameterized Algorithms, Algebraic Algorithms, Longest Cycle, Matroid Parity
Collection: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue Date: 2021
Date of publication: 02.07.2021

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