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

10/04/2017

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by   Holger Dell, et al.

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Focke, Goldberg, and Živný (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 Lovász (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 Lovász's framework to prove a complexity dichotomy for counting surjective homomorphisms to fixed finite structures.