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

Jin-Yi Cai; Zhiguo Fu; Shuai Shao

doi:10.1016/j.ic.2020.104589

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arXiv · 1904.02362

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

Jin-Yi Cai·Zhiguo Fu·Shuai Shao

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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.

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Jin-Yi Cai, Zhiguo Fu, Shuai Shao. 2019-04-04. Beyond #CSP: A Dichotomy for Counting Weighted Eulerian Orientations with ARS. https://doi.org/10.1016/j.ic.2020.104589

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