An Algorithmic Method of Partial Derivatives

TL;DR

This paper introduces an algorithmic framework for computing derivatives of polynomials applied to determinants, leading to faster algorithms for several combinatorial and algebraic problems, and explores new algebraic complexity measures related to matrix multiplication.

Contribution

The paper presents a novel algorithmic approach for partial derivatives of polynomials applied to determinants, improving complexity bounds for multiple problems and raising new questions in algebraic complexity theory.

Findings

Faster parameterized algorithms for matroid problems.

Deterministic algorithms for Edmonds's problem and subgraph detection.

New algebraic complexity measures related to matrix multiplication.

Abstract

We study the following problem and its applications: given a homogeneous degree-$d$ polynomial $g$ as an arithmetic circuit, and a $d \times d$ matrix $X$ whose entries are homogeneous linear polynomials, compute $g(\partial/\partial x_1, \ldots, \partial/\partial x_n) \det X$. 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 \emph{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 $\omega$. 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 $O(n 2^{\omega n})$, whereas all known upper bounds on the Waring rank of this polynomial exceed $n!$.