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.

翻译：自Valiant的开创性工作以来，图同态计数及其泛化问题——如计数约束满足问题（CSP）、其变体以及一般计数问题——得到了深入研究。虽然图同态精确计数（Dyer and Greenhill, 2000）与计数CSP（Bulatov, 2013; Dyer and Richerby, 2013）的复杂性已较为明确，但模某个自然数的计数问题同样引起了广泛关注。Faben与Jerrum在2015年的论文中提出一个猜想：对于固定图H模素数p的计数问题，除非H具有特定类型的自同构，否则在精确计数困难的情况下该模计数问题也是困难的。本文证实了这一猜想。在此研究过程中，我们发展了一系列技术，这些技术拓宽了模计数可用的归约方法谱系，并适用于一般CSP而非仅限于图同态问题。