2026-07-22T10:25:01Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14107 2024-03-06T10:53:29Z ddc:004 open_access

Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials Brand, Cornelius Pratt, Kevin Parameterized Algorithms Algebraic Algorithms Longest Cycle Matroid Parity 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cornelius Brand and Kevin Pratt 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.38 urn:nbn:de:0030-drops-141079 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.38 eng https://creativecommons.org/licenses/by/4.0/legalcode