  Hypertree decompositions, as well as the more powerful generalized hypertree
decompositions (GHDs), and the yet more general fractional hypertree
decompositions (FHD) are hypergraph decomposition methods successfully used for
answering conjunctive queries and for the solution of constraint satisfaction
problems. Every hypergraph H has a width relative to each of these
decomposition methods: its hypertree width hw(H), its generalized hypertree
width ghw(H), and its fractional hypertree width fhw(H), respectively.
  It is known that hw(H) <= k can be checked in polynomial time for fixed k,
while checking ghw(H) <= k is NP-complete for any k greater than or equal to 3.
The complexity of checking fhw(H) <= k for a fixed k has been open for more
than a decade.
  We settle this open problem by showing that checking fhw(H) <= k is
NP-complete, even for k=2. The same construction allows us to prove also the
NP-completeness of checking ghw(H) <= k for k=2. After proving these hardness
results, we identify meaningful restrictions, for which checking for bounded
ghw or fhw becomes tractable.
