Symmetric Algebraic Circuits and Homomorphism Polynomials
This work advances algebraic complexity theory by providing a general framework for symmetric circuits, with implications for understanding computational hardness in symmetric settings.
The authors tackled the problem of characterizing which symmetric polynomials admit small symmetric algebraic circuits, showing they are exactly those expressible as linear combinations of homomorphism counting polynomials of bounded-treewidth graphs, and they unconditionally established a known conditional dichotomy for immanant families in the symmetric setting.
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.