Complexity of #Hom for bounded adaptive width and unbounded fractional hypertree width

Ascertain the computational complexity of counting homomorphisms #Hom(H,G) when the pattern hypergraph family H has bounded adaptive width but unbounded fractional hypertree width; specifically, determine whether #Hom(H,G) is polynomial-time solvable (or fixed-parameter tractable) or instead computationally hard under standard assumptions.

Background

Within unbounded-rank hypergraphs, Grohe and Marx gave a polynomial-time algorithm for bounded fractional hypertree width, while Marx proved hardness for unbounded adaptive width. The remaining regime—bounded adaptive width but unbounded fractional hypertree width—has not been classified.

The paper explicitly points out this gap as still unknown, emphasizing that resolving it would close the remaining case in the known landscape of homomorphism counting complexity for unbounded-rank hypergraphs.