Binary Determinantal Complexity

TL;DR

This paper establishes a lower bound of 7x7 for representing the 3x3 permanent as a determinant with entries limited to zeros, ones, and variables, using a computer-assisted proof involving bipartite graph enumeration.

Contribution

It provides a novel computer-based proof technique for determinantal complexity and characterizes sequences of such determinants within a specific complexity class.

Findings

Proves 7x7 matrix size is necessary for the 3x3 permanent representation.

Characterizes sequences of determinants with zeros, ones, and variables as constant free weakly skew circuits.

Uses bipartite graph enumeration in the proof process.

Abstract

We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. We show that these are exactly the sequences in the complexity class of constant free polynomially sized (weakly) skew circuits.