Complexity Classification of Counting Graph Homomorphisms Modulo a Prime Number

SIAM Journal on Computing, Ahead of Print.
Abstract. Counting graph homomorphisms and its generalizations such as the counting constraint satisfaction problem (CSP), variations of the counting CSP, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms [M. Dyer and C. Greenhill, Random Structures Algorithms, 17 (2000), pp. 260–289] and the counting CSP [A. A. Bulatov, J. ACM, 60 (2013), pp. 34:1–34:41, and M. E. Dyer and D. Richerby, SIAM J. Comput., 42 (2013), pp. 1245–1274] is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper, [J. Faben and M. Jerrum, Theory Comput., 11 (2015), pp. 35–57] suggested a conjecture stating that counting homomorphisms to a fixed graph [math] modulo a prime number is hard whenever it is hard to count exactly unless [math] 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.