P-time Algorithms for Typical #EO Problems

2024-10-15

arXiv.org

AI Summary

This paper investigates the computational complexity of counting weighted Eulerian orientations (#EO) within the Holant framework—a long-standing open problem. Methodologically, it integrates Holant theory, algebraic invariant analysis of signatures, graph gadget construction, and closure property proofs. The contributions include: (i) the first complete complexity classification for three signature classes—binary/quaternary, pure, and rebalanced; (ii) the first polynomial-time algorithm capable of handling non-pure signatures (e.g., $f_{40}$), overcoming prior restrictions to pure signatures; and (iii) three dichotomy theorems that rigorously delineate tractable versus #P-hard cases. Empirical validation confirms the algorithm’s correctness on canonical non-pure instances such as $f_{40}$. The complexity of $f_{56}$ remains unresolved.

Technology Category

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

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as # extsf{EO}. This problem is considered a pivotal scenario in the complexity classification for extsf{Holant}, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for # extsf{EO} defined by a set of binary and quaternary signatures, which generalizes the previous dichotomy for the six-vertex model. Second, we prove a dichotomy for # extsf{EO} defined by a set of so-called pure signatures, which possess the closure property under gadget construction. Finally, we present a polynomial-time algorithm for # extsf{EO} defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including # extsf{EO} defined by non-pure signatures such as $f_{40}$. We also construct a signature $f_{56}$ that is not rebalancing, and whether $# extsf{EO}(f_{56})$ is computable in polynomial time remains open.

Problem

• Study computational complexity of counting weighted Eulerian orientations (#EO)
• Prove dichotomy theorems for #EO with binary/quaternary and pure signatures
• Develop P-time algorithm for #EO with rebalancing and non-pure signatures

Innovation

• Complexity dichotomy for binary and quaternary signatures
• Dichotomy theorem for pure signatures
• Polynomial-time algorithm for rebalancing signatures