On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

TL;DR

This paper investigates the complexity of counting graph homomorphisms over finite fields of prime order, introducing quantum graphs, reducing problems to bipartite graphs, and establishing a comprehensive complexity dichotomy.

Contribution

It introduces quantum graphs into modular counting homomorphisms, reduces complexity analysis to bipartite graphs, and provides a complete dichotomy for certain bipartite graph classes.

Findings

Complexity collapses to classical graph criteria at dimension 1.

Reduces problem complexity analysis to bipartite graphs.

Establishes a complete dichotomy for specific bipartite graph classes.

Abstract

We study the problem of counting the number of homomorphisms from an input graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and Jerrum [ToC'15] and its complexity is subject to a growing series of research articles, e.g. the work of Focke, Goldberg, Roth, and Zivn\'y [SIDMA'21] and the work of Bulatov and Kazeminia [STOC'22], subsequent to this article's conference version. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph $\bar{H}$ collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite ($K_{3,3}\backslash\{e\}$, ${domino}$)-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with $p$ equal to $2$, this establishes new results.