General and Fractional Hypertree Decompositions: Hard and Easy Cases

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.