# Beyond #CSP: A Dichotomy for Counting Weighted Eulerian Orientations with ARS

## TL;DR

This paper classifies the computational complexity of counting weighted Eulerian orientations with arrow reversal symmetry, establishing a dichotomy that they are either polynomial-time solvable or #P-hard, and reveals a deep connection with weighted #CSP problems.

## Contribution

It introduces a new prime factorization framework for constraint functions and proves a complexity dichotomy for these orientation problems, linking them to weighted #CSPs through structural analysis.

## Key findings

- All such problems are either polynomial-time computable or #P-hard.
- Weighted Eulerian orientation problems encompass all weighted Boolean #CSPs.
- A novel structural connection between #CSP and Eulerian orientations is established.

## Abstract

We define and explore a notion of unique prime factorization for constraint functions, and use this as a new tool to prove a complexity classification for counting weighted Eulerian orientation problems with arrow reversal symmetry (ARS). We prove that all such problems are either polynomial-time computable or #P-hard. We show that the class of weighted Eulerian orientation problems subsumes all weighted counting constraint satisfaction problems (#CSP) on Boolean variables. More significantly, we establish a novel connection between #CSP and counting weighted Eulerian orientation problems that is global in nature. This connection is based on a structural determination of all half-weighted affine linear subspaces over $\mathbb{Z}_2$, which is proved using M\"obius inversion.