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

2025-02-04

AI Summary

This paper resolves the long-standing open problem of the exact complexity classification of counting weighted Eulerian orientations (#EO). Within the Holant framework, it establishes the first dichotomy theorem for #EO, proving that every instance is either #P-complete or solvable in polynomial time with an NP oracle—i.e., FP^NP-complete. The proof integrates algebraic analysis, modular reduction, lattice theory, and symmetry characterization under constraint satisfaction, all anchored in the Holant signature reduction framework. This result fully characterizes the computational complexity of #EO and directly yields tight dichotomies for three important Holant subclasses. Moreover, it provides a critical breakthrough and foundational support toward resolving the overarching Holant dichotomy conjecture.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

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 $ ext{FP}^ ext{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.

Problem

• Resolving Holant complexity classification
• Dichotomy for #EO complexity
• Establishing #P-hard vs. FP^NP

Innovation

• FP^NP vs. #P dichotomy
• Complexity classification for #EO
• Polynomial-time computable with NP oracle