2025/02/04 by Meng, Boning, Wang, Juqiu, Xia, Mingji · 2 citations
#Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2502.02012
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 FPNP 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.