On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials

TL;DR

This paper develops explicit algebraic branching programs for rectangular determinant and permanent polynomials, providing new size bounds in both commutative and noncommutative settings, and establishes hardness results for noncommutative determinant evaluation.

Contribution

It constructs explicit ABPs of specific sizes for rectangular determinant and permanent polynomials, advancing understanding of their algebraic circuit complexity.

Findings

Explicit ABP of size O*(n choose k/2) for noncommutative permanent.

Explicit ABP of size O*(2^k) for commutative rectangular determinant.

Noncommutative determinant evaluation is W[1]-hard over rationals.

Abstract

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the \emph{rectangular} symbolic matrix in both commutative and noncommutative settings. The main results are: 1. We show an explicit $O^{*}({n\choose {\downarrow k/2}})$-size ABP construction for noncommutative permanent polynomial of $k\times n$ symbolic matrix. We obtain this via an explicit ABP construction of size $O^{*}({n\choose {\downarrow k/2}})$ for $S_{n,k}^*$, noncommutative symmetrized version of the elementary symmetric polynomial $S_{n,k}$. 2. We obtain an explicit $O^{*}(2^k)$-size ABP construction for the commutative rectangular determinant polynomial of the $k\times n$ symbolic matrix. 3. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is $W[1]$-hard.