An Algorithmic Method of Partial Derivatives
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We study the following problem and its applications: given a homogeneous degree-d polynomial g as an arithmetic circuit, and a d × d matrix X whose entries are homogeneous linear polynomials, compute g(∂/∂ x_1, …, ∂/∂ x_n) X. By considering special cases of this problem we obtain faster parameterized algorithms for several problems, including the matroid k-parity and k-matroid intersection problems, faster deterministic algorithms for testing if a linear space of matrices contains an invertible matrix (Edmonds's problem) and detecting k-internal outbranchings, and more. We also match the runtime of the fastest known deterministic algorithm for detecting subgraphs of bounded pathwidth, while using a new approach. Our approach raises questions in algebraic complexity related to Waring rank and the exponent of matrix multiplication ω. In particular, we study a new complexity measure on the space of homogeneous polynomials, namely the bilinear complexity of a polynomial's apolar algebra. Our algorithmic improvements are reflective of the fact that for the degree-n determinant polynomial this quantity is at most O(n 2^ω n), whereas all known upper bounds on the Waring rank of this polynomial exceed n!.
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