Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials

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, . . . , ∂/∂xn) det X. We show that this quantity can be computed using 2|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|C| poly(n, d) operations, and if furthermore X is a Hankel matrix, to φ2d|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]. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms