On the complexity of the permanent in various computational models

TL;DR

This paper investigates the computational complexity of the permanent and determinant, providing bounds on their determinantal complexity and establishing limitations on rank-one determinantal expressions, while also exploring various computational models.

Contribution

It establishes an O(m^3) bound for the regular determinantal complexity of the determinant and proves the non-existence of rank-one determinantal expressions for perm_m and det_m when m ≥ 3, extending previous results.

Findings

Determinantal complexity of det_m is O(m^3)

No rank-one determinantal expression exists for perm_m or det_m when m ≥ 3

Several folklore results relating different computational models are proved

Abstract

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation.