The Converse of the Real Orthogonal Holant Theorem
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911156838334464 |
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| author | Young, Ben |
| author_facet | Young, Ben |
| contents | The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. 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 simultaneosly orthogonally decomposable (odeco) sets of symmetric tensors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06911 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Converse of the Real Orthogonal Holant Theorem Young, Ben Discrete Mathematics Combinatorics G.2.1 The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. 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 simultaneosly orthogonally decomposable (odeco) sets of symmetric tensors. |
| title | The Converse of the Real Orthogonal Holant Theorem |
| topic | Discrete Mathematics Combinatorics G.2.1 |
| url | https://arxiv.org/abs/2409.06911 |