The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO
Computational Complexity
2025-04-28 v2
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 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.
Cite
@article{arxiv.2502.02012,
title = {The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO},
author = {Boning Meng and Juqiu Wang and Mingji Xia},
journal= {arXiv preprint arXiv:2502.02012},
year = {2025}
}