P-time Algorithms for Typical #EO Problems

TL;DR

This paper investigates the computational complexity of counting weighted Eulerian orientations within the Holant framework, establishing a dichotomy theorem and providing a polynomial-time algorithm for certain cases, advancing understanding of these counting problems.

Contribution

It introduces a complexity dichotomy for #EO with binary and quaternary signatures, and extends polynomial-time algorithms to broader classes including non-pure signatures.

Findings

Proved a complexity dichotomy theorem for #EO with specific signatures.

Established a polynomial-time algorithm for #EO with rebalancing signatures.

Identified an open problem regarding the polynomial-time computability of #EO with signature f_{56}.

Abstract

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as \#\textsf{EO}. This problem is considered a pivotal scenario in the complexity classification for \textsf{Holant}, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for \#\textsf{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 \#\textsf{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 \#\textsf{EO} defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including \#\textsf{EO} defined by non-pure signatures such as $f_{40}$. We also construct a signature $f_{56}$ that is not rebalancing, and whether $\#\textsf{EO}(f_{56})$ is computable in polynomial time remains open.