Efficiently bounding cardinality ratios through database constraints

Numerical dependencies (NDs) are a type of database constraints in which one limits the number of distinct Y-values that can appear together with any X-value, where both X and Y are sets of attributes. The seminal work by Grant and Minker has shown that NDs are not finitely axiomatizable, which has cut further investigation on this kind of constraints. In this paper we show that, given a set of sound inference rules similar to those used for functional dependencies, the membership problem for NDs is NP-hard, and propose a branch & bound algorithm for efficiently solving the problem. The algorithms adopts a suite of optimization strategies that make it applicable in practice, providing considerable speed-up over a naive approach.