A combinatorial view of Holant problems on higher domains

Yin Liu

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arXiv · 2403.14150

A combinatorial view of Holant problems on higher domains

Yin Liu

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Abstract

On the Boolean domain, there is a class of symmetric signatures called ``Fibonacci gates'' for which a beautiful P-time combinatorial algorithm has been designed for the corresponding $\operatorname{Holant}$ problems. In this work, I give a combinatorial view for $\operatorname{Holant}(\mathcal{F})$ problems on a domain of size 3 where $\mathcal{F}$ is a set of arity 3 functions with inputs taking values on the domain of size 3 and the functions share some common properties. The combinatorial view can also be extended to the domain of size 4. Specifically, I extend the definition of "Fibonacci gates" to the domain of size 3 and the domain of size 4. Moreover, I give the corresponding combinatorial algorithms.

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Yin Liu. 2024-03-21. A combinatorial view of Holant problems on higher domains. https://arxiv.org/abs/2403.14150

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