🤖 AI Summary
This work resolves the Faben–Jerrum conjecture, which posits that the computational complexity of counting graph homomorphisms modulo a prime $p$ coincides with that of exact counting, unless the target graph admits certain nontrivial automorphisms. Leveraging a synthesis of algebraic graph theory, group action analysis, and modular arithmetic reduction techniques, we provide the first complete proof of the conjecture and extend the modular counting reduction framework from graph homomorphisms to general constraint satisfaction problems (#$_p$CSP). Our main contributions are: (1) a full dichotomy classification for modular graph homomorphism counting—establishing completeness for either $mathsf{P}$ or $#mathsf{_pP}$; (2) the first generic reduction method applicable to arbitrary #$_p$CSPs; and (3) a precise characterization showing that automorphism structure fundamentally governs tractability in modular counting. These results furnish foundational tools and a systematic complexity-theoretic characterization for modular counting.
📝 Abstract
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.