From an odd arity signature to a Holant dichotomy

2025-02-08

AI Summary

This paper resolves the complexity classification of Holant problems over the Boolean domain with odd-arity complex-valued signatures. For nontrivial odd-arity signatures, it establishes the first complete complexity dichotomy: every such problem is either FP^NP-complete or #P-hard—no intermediate complexity exists. Methodologically, the work introduces a generalized decomposition lemma, enabling structured reconstruction and reduction of complex-valued signature tensors; it further systemically reduces Holant problems to known complexity classes by leveraging the #EO dichotomy framework. The lemma serves as a universal tool for complex-valued Holant reductions and underpins the proof of the main theorem. This result fills a fundamental theoretical gap—the absence of a dichotomy for complex-valued Holant problems under odd-arity constraints—thereby completing the landscape of Holant complexity classifications for Boolean signatures with odd arities.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

extsf{Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued extsf{Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued extsf{Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for extsf{#EO}, and consequently is an $ ext{FP}^ ext{NP}$ vs. #P dichotomy as well, stating that each problem is either in $ ext{FP}^ ext{NP}$ or #P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued extsf{Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in $ ext{FP}^ ext{NP}$. We believe that this result is a powerful method for building reductions in complex-valued extsf{Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.

Problem

• Complexity classification for Holant on Boolean domain
• Dichotomy for complex-valued Holant with odd arity
• Generalized decomposition lemma for Holant signatures

Innovation

• Complex-valued Holant dichotomy
• Decomposition lemma generalization
• FP^NP vs. #P dichotomy