The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO

Boning Meng; Juqiu Wang; Mingji Xia

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arXiv · 2502.02012

The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO

Boning Meng·Juqiu Wang·Mingji Xia

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Abstract

The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as #EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an $\text{FP}^\text{NP}$ vs. #P dichotomy for #EO, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for #EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for #EO.

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Boning Meng, Juqiu Wang, Mingji Xia. 2025-02-04. The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO. https://arxiv.org/abs/2502.02012

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