Complexity Analysis of Generalized and Fractional Hypertree Decompositions

Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph $H$ has a width relative to each of these methods: its hypertree width $hw(H)$, its generalized hypertree width $ghw(H)$, and its fractional hypertree width $fhw(H)$, respectively. It is known that $hw(H)\leq k$ can be checked in polynomial time for fixed $k$, while checking $ghw(H)\leq k$ is NP-complete for $k \geq 3$. The complexity of checking $fhw(H)\leq k$ for a fixed $k$ has been open for over a decade. We settle this open problem by showing that checking $fhw(H)\leq k$ is NP-complete, even for $k=2$. The same construction allows us to prove also the NP-completeness of checking $ghw(H)\leq k$ for $k=2$. After that, we identify meaningful restrictions which make checking for bounded $ghw$ or $fhw$ tractable or allow for an efficient approximation of the $fhw$.