Symmetric Algebraic Circuits and Homomorphism Polynomials

2025-02-10

AI Summary

This paper develops symmetric algebraic complexity theory, addressing the central question: when can a family of symmetric polynomials be computed by small symmetric algebraic circuits? The authors establish the first exact correspondence between symmetric circuit complexity and graph structural parameters—namely, treewidth and vertex cover number—proving that a symmetric polynomial is computable by a small symmetric circuit if and only if it admits a linear combination representation of homomorphism-counting polynomials over graphs of bounded treewidth. This transforms conditional classifications into unconditional characterizations, fully resolving the symmetric complexity of subgraph-counting polynomials. It further yields the first unconditional dichotomy for the immanant family under symmetric computation and rigorously establishes exponential lower bounds for the permanent and related functions in the symmetric circuit model.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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) 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.

Problem

• Symmetric algebraic circuits complexity
• Homomorphism polynomials representation
• Immanant families symmetric complexity

Innovation

• symmetric algebraic circuits
• homomorphism counting polynomials
• bounded treewidth graphs