On the Principal Minor Expansion and Complexity of the Symmetrized Determinant

TL;DR

This paper investigates the algebraic properties and computational complexity of the symmetrized determinant, revealing its connections to algebraic complexity classes and its computational hardness.

Contribution

It establishes that the symmetrized determinant's principal minor expansion resembles the ordinary determinant and proves its computational hardness and completeness in algebraic complexity classes.

Findings

Principal minor expansion of the symmetrized determinant is analogous to the ordinary determinant.

Computing the symmetrized determinant is -hard for some algebra.

The associated polynomial family is -complete over certain algebras.

Abstract

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.