Counting Homomorphisms to Trees Modulo a Prime

TL;DR

This paper establishes a clear complexity dichotomy for counting graph homomorphisms to trees modulo any prime, showing they are either efficiently computable or #_pP-complete, extending previous results from modulo 2 to all primes.

Contribution

It proves that for every tree and prime, counting homomorphisms modulo p is either polynomial-time or #_pP-complete, confirming a conjecture for all primes.

Findings

Dichotomy for homomorphism counting to trees modulo p established

Weighted independent set counting modulo p also classified

Results extend modular counting dichotomies from prime 2 to all primes

Abstract

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular, independent sets and colourings. In this article we study the complexity of $\#_p\mathsf{H}$OMS$\mathsf{T}$O$H$, the problem of counting graph homomorphisms from an input graph to a graph $H$ modulo a prime number $p$. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the input graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. However, in subsequent studies on counting modulo $2$ the influence, the structure of $H$ has on the tractability, was shown to persist, yielding similar dichotomies. Our main result shows that for every tree $H$ and every prime $p$ the problem $\#_p\mathsf{H}$OMS$\mathsf{T}$O$H$ is either polynomial time computable or $\#_p\mathsf{P}$-complete. This addresses the conjecture of Faben and Jerrum stating this dichotomy for every graph $H$ when counting modulo 2. In order to prove this result, we study the structural properties of a homomorphism. As an important interim, this study yields a dichotomy for the problem of weighted counting independent sets in a bipartite graph modulo some prime $p$. Our results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo~2 case but for the modular counting functions of all primes $p$.