CTD characterisation of hypertree width

Determine whether, for every hypergraph H and integer k ≥ 1, there exists a set of candidate bags S_{H,k} such that H admits a component-normal-form candidate tree decomposition using only bags from S_{H,k} if and only if the hypertree width of H is at most k.

Background

Candidate tree decompositions (CTDs) provide a unifying framework to compute various decomposition notions by fixing a set of allowable bags and searching for a tree decomposition using only those bags. Prior work showed how to reduce the recognition of bounded generalized and fractional hypertree width to CTDs for suitable candidate sets, yielding tractable fragments. However, hypertree decompositions impose the special condition involving parent/child relationships, which complicates a direct CTD-based characterisation.

The paper introduces soft hypertree width specifically to circumvent this difficulty, but explicitly notes that it is not known whether an exact CTD characterisation exists for hypertree width itself. Resolving this would clarify whether hypertree width can be captured within the CTD framework without relaxation.