Note on "The Complexity of Counting Surjective Homomorphisms and Compactions"

TL;DR

This paper simplifies the proof of #P-hardness for counting surjective homomorphisms from graphs with loops to fixed graphs, extending previous results using Lovász's framework.

Contribution

It provides a shorter proof of #P-hardness for a more general problem, applying Lovász's framework and building on prior work by Curticapean, Dell, Marx, and Chen.

Findings

Proves #P-hardness for counting surjective homomorphisms with loops in the input graph.

Uses Lovász's framework to simplify the proof approach.

Extends previous complexity results to more general graph classes.

Abstract

Focke, Goldberg, and \v{Z}ivn\'y (arXiv 2017) prove a complexity dichotomy for the problem of counting surjective homomorphisms from a large input graph G without loops to a fixed graph H that may have loops. In this note, we give a short proof of a weaker result: Namely, we only prove the #P-hardness of the more general problem in which G may have loops. Our proof is an application of a powerful framework of Lov\'asz (2012), and it is analogous to proofs of Curticapean, Dell, and Marx (STOC 2017) who studied the "dual" problem in which the pattern graph G is small and the host graph H is the input. Independently, Chen (arXiv 2017) used Lov\'asz's framework to prove a complexity dichotomy for counting surjective homomorphisms to fixed finite structures.