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Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Authors Marek Černý , Tim Seppelt

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LIPIcs.STACS.2026.25.pdf
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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Finite Model Theory
Keywords
  • treewidth
  • Courcelle’s theorem
  • logspace
  • multiplicity automata
  • polynomial identity testing

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Abstract

Two graphs G and H are homomorphism indistinguishable over a graph class ℱ if they admit the same number of homomorphisms from every graph F ∈ ℱ. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems HomInd(ℱ) of deciding homomorphism indistinguishability over ℱ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of HomInd(ℱ), Seppelt (MFCS 2024) showed that HomInd(ℱ) is in randomised polynomial time for every graph class ℱ of bounded treewidth that can be defined in counting monadic second-order logic CMSO₂.
We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in P. For CMSO₂-definable graph classes ℱ of bounded pathwidth, we improve the previous complexity upper bound for HomInd(ℱ) from P to C_ = L and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as C_ = L-complete.

Cite As Get BibTex

Marek Černý and Tim Seppelt. Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.25

Author Details

Marek Černý
  • Universiteit Antwerpen, Belgium
Tim Seppelt
  • IT-Universitetet i København, Denmark

Funding

  • Černý, Marek: University of Antwerp (BOF, Doctoral project 47103).
  • 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 Mikołaj Bojańczyk and Sam van Gool at the Dagstuhl Seminar 25141 "Categories for Automata and Language Theory". Furthermore, we are grateful for discussions with David E. Roberson and Louis Härtel. Finally, we gratefully acknowledge Floris Geerts for drawing our attention to the link between automata and Specht-Wiegmann-type theorems.

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