Observations on Symmetric Circuits

TL;DR

This paper investigates symmetric arithmetic circuits, providing simplified proofs and extending exponential lower bounds for the permanent and related polynomials, emphasizing the significance of group symmetry in computational complexity.

Contribution

It offers a simpler proof of exponential lower bounds for the permanent in symmetric circuits and extends these bounds to a broad class of polynomials, highlighting the importance of group structure.

Findings

Exponential lower bounds for the permanent in symmetric circuits.

Large class of polynomials also require exponential size.

Super-polynomial bounds hold for smaller symmetry groups.

Abstract

We study symmetric arithmetic circuits and improve on lower bounds given by Dawar and Wilsenach (ArXiv 2020). Their result showed an exponential lower bound of the permanent computed by symmetric circuits. We extend this result to show a simpler proof of the permanent lower bound and show that a large class of polynomials have exponential lower bounds in this model. In fact, we prove that all polynomials that contain at least one monomial of the permanent have exponential size lower bounds in the symmetric computation model. We also show super-polynomial lower bounds for smaller groups. We support our conclusion that the group is much more important than the polynomial by showing that on a random process of choosing polynomials, the probability of not encountering a super-polynomial lower bound is exponentially low.