Holant* Dichotomy on Domain Size 3: A Geometric Perspective
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909584737697792 |
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| author | Cai, Jin-Yi Ihm, Jin Soo |
| author_facet | Cai, Jin-Yi Ihm, Jin Soo |
| contents | Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holant* Dichotomy on Domain Size 3: A Geometric Perspective Cai, Jin-Yi Ihm, Jin Soo Computational Complexity Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy. |
| title | Holant* Dichotomy on Domain Size 3: A Geometric Perspective |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2504.14074 |