The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability on the other hand is algebraic and based on properties of polynomials.

中文翻译：

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非负权重有向图同态的可判定二分定理

图同态问题的复杂性多年来一直是深入研究的主题。在本文中，我们证明了有向图同态的配分函数的可判定复杂度二分定理。我们的定理适用于问题的所有非负加权形式：给定任何具有非负代数项的固定矩阵 A，来自任何有向图 G 的有向图同态的配分函数 ZA(G) 要么在多项式时间内易于处理，要么# P-hard，取决于矩阵 A。二分定理的证明是组合的，但涉及到图同态问题的无限族的定义。另一方面，其可判定性的证明是代数的，并且基于多项式的性质。