Bi-polynomial rank and determinantal complexity

TL;DR

This paper introduces the bi-polynomial rank, a new polynomial measure related to arithmetic branching programs, and demonstrates its use in improving lower bounds for the determinantal complexity of the permanent, offering a novel approach to the permanent vs. determinant problem.

Contribution

It defines the bi-polynomial rank, relates it to determinantal complexity, and proposes a new computational approach using concave minimization to advance the permanent vs. determinant problem.

Findings

Bi-polynomial rank provides a lower bound for determinantal complexity.

The lower bound for the permanent's determinantal complexity is improved to (d-1)^2 + 1.

A new approach using positive semidefiniteness and concave minimization is proposed.

Abstract

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d^2/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank concept of polynomials, called the bi-polynomial rank. The bi-polynomial rank is related to width of an arithmetic branching program. The bi-polynomial rank of a homogeneous polynomial p of even degree 2k is defined as the minimum n such that p can be written as a summation of n products of polynomials of degree k. We prove that the bi-polynomial rank gives a lower bound of the determinantal complexity. As a consequence, the above Mignon and Ressayre bound is improved to (d-1)^2 + 1 over the field of reals. We show that the computation of the bi-polynomial rank is formulated as a rank minimization problem. Applying the concave minimization technique, we reduce the problem of lower-bounding determinantal complexity to that of proving the positive semidefiniteness of matrices, and this is a new approach for the permanent vs. determinant problem. We propose a computational approach for giving a lower bound of this rank minimization, via techniques of the concave minimization. This also yields a new strategy to attack the permanent vs. determinant problem.