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The Converse of the Real Orthogonal Holant Theorem

Author Ben Young

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  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Holant
  • Counting Indistinguishability
  • Odeco

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Abstract

The Holant theorem is a powerful tool for studying the computational complexity of counting problems. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very powerful counting indistinguishability theorem. The most general converse does not hold, but we prove the following, still highly general, version: if any two sets of real-valued signatures are Holant-indistinguishable, then they are equivalent up to an orthogonal transformation. This resolves a partially open conjecture of Xia (2010). Consequences of this theorem include the well-known result that homomorphism counts from all graphs determine a graph up to isomorphism, the classical sufficient condition for simultaneous orthogonal similarity of sets of real matrices, and a combinatorial characterization of sets of simultaneosly orthogonally decomposable (odeco) symmetric tensors.

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Ben Young. The Converse of the Real Orthogonal Holant Theorem. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 136:1-136:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.136

Author Details

Ben Young
  • Department of Computer Sciences, University of Wisconsin-Madison, WI, USA

Acknowledgements

The author thanks Jin-Yi Cai for helpful discussions and Ashwin Maran for his suggestion which improved the proof of Theorem 27. The author also thanks Guus Regts for pointing out reference [Schrijver, 2008].

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