Complexity classification of counting graph homomorphisms modulo a prime number

TL;DR

This paper classifies the computational complexity of counting graph homomorphisms modulo a prime, confirming a conjecture that relates the difficulty of modular counting to exact counting, and introduces new reduction techniques for CSPs.

Contribution

It confirms a conjecture linking the hardness of modular counting to exact counting and develops new reduction techniques applicable to general CSPs.

Findings

Counting homomorphisms modulo a prime is hard if and only if exact counting is hard, for fixed graphs.

New reduction techniques extend the analysis from graph homomorphisms to general CSPs.

The conjecture by Faben and Jerrum is proven to be true.

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.