Complete dichotomy for counting hypergraph homomorphisms

Determine a complete complexity dichotomy for counting homomorphisms between general hypergraphs of unbounded rank: Given a family of pattern hypergraphs H, establish necessary and sufficient structural conditions on H that characterize exactly when the problem of computing #Hom(H,G) is solvable efficiently (for example, in polynomial or fixed-parameter tractable time) versus when it is computationally intractable.

Background

The paper recalls that for graphs, the complexity of counting homomorphisms is fully understood via treewidth, and for bounded-rank hypergraphs a dichotomy is known. In contrast, when the rank is unbounded, only partial results are available.

Specifically, prior work gives a polynomial-time algorithm for counting homomorphisms from hypergraphs of bounded fractional hypertree width and hardness for classes of unbounded adaptive width, but a unifying criterion that yields a full dichotomy over general hypergraphs is not known. The authors highlight this gap as a central open problem in the introduction, setting context for their own results on sub- and induced sub-hypergraph counting.