Counting graph homomorphisms and its generalizations such as the Counting Constraint Satisfaction Problem (CSP), its variations, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms (Dyer and Greenhill, 2000) and the counting CSP (Bulatov, 2013, and Dyer and Richerby, 2013) is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper Faben and Jerrum suggested a conjecture stating that counting homomorphisms to a fixed graph H modulo a prime number is hard whenever it is hard to count exactly, unless H has automorphisms of certain kind. In this paper we confirm this conjecture. As a part of this investigation we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.

中文翻译：

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以素数为模的计数图同态的复杂度分类

自 Valiant 的开创性工作以来，人们对计数图同态及其泛化（例如计数约束满足问题 (CSP)、其变体和一般计数问题）进行了深入研究。虽然图同态的精确计数（Dyer 和 Greenhill，2000）和计数 CSP（Bulatov，2013 和 Dyer 和 Richerby，2013）的复杂性很好理解，但对某个自然数取模的计数也引起了相当大的兴趣。在他们 2015 年的论文 Faben 和 Jerrum 中，提出了一个猜想，指出在难以准确计数的情况下，对固定图 H 模质素数的同态进行计数是很困难的，除非 H 具有某种类型的自同构。在本文中，我们证实了这一猜想。