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Symmetric Algebraic Circuits and Homomorphism Polynomials

Anuj Dawar, Benedikt Pago, Tim Seppelt

TL;DR

This work develops a general theory of symmetric algebraic circuits by linking the symmetric circuit complexity of matrix-symmetric polynomials to graph-homomorphism counting polynomials. It proves a central characterisation: a family of matrix-symmetric polynomials has small symmetric circuits (in terms of orbit size) iff it can be written as a linear combination of homomorphism polynomials of graphs with bounded treewidth. The authors further relate this to counting width, showing bounded counting width implies tractable symmetric circuits in several restricted cases, and use this to obtain unconditional dichotomies for immanants. The results unify and extend prior lower-bound strategies for symmetric computation, connect circuit complexity with motif counting in graphs, and open avenues for applying these methods to subgraph polynomials, vertex-cover-type parameters, and potential proof-system lower bounds. Overall, the paper provides a broad, non-uniform, symmetry-aware framework that clarifies when symmetric circuits are powerful and when they must incur super-polynomial size.

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.

Symmetric Algebraic Circuits and Homomorphism Polynomials

TL;DR

This work develops a general theory of symmetric algebraic circuits by linking the symmetric circuit complexity of matrix-symmetric polynomials to graph-homomorphism counting polynomials. It proves a central characterisation: a family of matrix-symmetric polynomials has small symmetric circuits (in terms of orbit size) iff it can be written as a linear combination of homomorphism polynomials of graphs with bounded treewidth. The authors further relate this to counting width, showing bounded counting width implies tractable symmetric circuits in several restricted cases, and use this to obtain unconditional dichotomies for immanants. The results unify and extend prior lower-bound strategies for symmetric computation, connect circuit complexity with motif counting in graphs, and open avenues for applying these methods to subgraph polynomials, vertex-cover-type parameters, and potential proof-system lower bounds. Overall, the paper provides a broad, non-uniform, symmetry-aware framework that clarifies when symmetric circuits are powerful and when they must incur super-polynomial size.

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.

Paper Structure

This paper contains 52 sections, 68 theorems, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Characterising matrix-symmetric polynomials admitting small symmetric circuits.
  3. Lower bounds via counting width.
  4. Related work.
  5. Preliminaries
  6. Permutation groups.
  7. Symmetric circuits.
  8. Complexity measures for symmetric circuits.
  9. Counting logic.
  10. Overview of results
  11. Characterisation of symmetric circuit complexity by homomorphism polynomials
  12. Symmetric complexity of homomorphism polynomials
  13. Symmetric complexity of subgraph polynomials
  14. Symmetric complexity of the immanants
  15. Extended Preliminaries
  16. ...and 37 more sections

Key Result

Theorem 1.1

For every family of polynomials $p_{n,m} \in \mathbb{Q}[\mathcal{X}_{n,m}]$, the following are equivalent:

Figures (2)

  • Figure 1: Implications between properties of families of matrix-symmetric polynomials $(p_{n,m})$. The dashed implications hold under additional assumptions.
  • Figure 2: The gadget that replaces an edge $uv$.

Theorems & Definitions (160)

  • Theorem 1.1: restate=thmMain,label=thm:main1,name=
  • Lemma 2.1: restate=constantSupportOfGates,label=lem:constantSupportOfGates,name=
  • Lemma 2.2: restate=constantSupportImpliesPolyOrbit,label=lem:constantSupportImpliesPolyOrbit,name=
  • Lemma 3.0
  • proof : Proof sketch of \ref{['thm:main1']}
  • Theorem 3.1: restate=hardPolynomialsInT,label=thm:hardPolynomialsInT,name=
  • Definition 3.2: anderson_symmetric_2017
  • Theorem 3.2
  • Theorem 3.3: see \ref{['thm:lincomb']}
  • Theorem 3.4: see \ref{['thm:sublinear']}
  • ...and 150 more