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Symmetric Algebraic Circuits and Homomorphism Polynomials

Authors Anuj Dawar , Benedikt Pago , Tim Seppelt

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LIPIcs.ITCS.2026.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • algebraic complexity
  • finite model theory
  • symmetric circuits
  • homomorphism counting
  • graph homomorphism
  • treewidth
  • counting width
  • first-order logic with counting quantifiers

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Abstract

The central open question of algebraic complexity is whether VP ≠ 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.

Cite As Get BibTex

Anuj Dawar, Benedikt Pago, and Tim Seppelt. Symmetric Algebraic Circuits and Homomorphism Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.46

Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Benedikt Pago
  • Department of Computer Science and Technology, University of Cambridge, UK
Tim Seppelt
  • IT University Copenhagen, Denmark

Funding

The first and second author were funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee: grant number EP/X028259/1.
  • Seppelt, Tim: European Union (CountHom, 101077083). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We would like to acknowledge fruitful discussions with Radu Curticapean, Filip Kučerák, Deepanshu Kush, and Benjamin Rossman.

References

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