Symmetric Algebraic Circuits and Homomorphism Polynomials

TL;DR

This paper develops a symmetric algebraic complexity theory linking symmetric circuit size to graph homomorphism polynomials and explores the symmetric complexity of immanants, advancing understanding of algebraic circuit lower bounds.

Contribution

It establishes a characterization of symmetric polynomials with small circuits via homomorphism polynomials and relates symmetric complexity to graph parameters, extending prior lower bound results.

Findings

Symmetric polynomial complexity characterized by homomorphism polynomials of bounded treewidth graphs.

Unconditional dichotomy for symmetric complexity of immanants, previously conditional.

Relationship between symmetric complexity and graph parameters like vertex cover number.

Abstract

The central open question of algebraic complexity is whether VP is unequal to VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been confirmed by Dawar and Wilsenach (2020, 2025) who showed exponential lower bounds on the size of symmetric circuits for the permanent. In this work, we set out to develop a more general symmetric algebraic complexity theory. Our main result is that a family of symmetric polynomials admits small symmetric circuits if and only if they can be written as a linear combination of homomorphism counting polynomials of graphs of bounded treewidth. We also establish a relationship between the symmetric complexity of subgraph counting polynomials and the vertex cover number of the pattern graph. As a concrete example, we examine the symmetric complexity of immanant families (a generalisation of the determinant and permanent) and show that a known conditional dichotomy due to Curticapean (2021) holds unconditionally in the symmetric setting.