On the relationship of minimum and optimum covers for a set of functional dependencies

Most algorithms in relational database theory use a set of functional dependencies as their input. The efficiency of the algorithms depends on the size of the set. The notions of a minimum set (with as few dependencies as possible) and an optimum set (which is as short as possible) were introduced by Maier. He showed that while a minimum cover for a given set of dependencies can be found in polynomial time, obtaining an optimum cover is an NP-complete problem. Here the relationship of these covers is explored further. It is shown that the length of a minimum set (i) cannot be bounded by a linear function on the length of an optimum cover, and (ii) is bounded by the square of the length of an optimum cover. It is also shown that the NP-completeness of the optimization problem is somewhat surprisingly caused solely by the difficulty of optimizing a single class of dependencies having equivalent left sides, not by the globality of the optimality condition. This result has some practical significance, since the equivalence classes appearing in practice are short. The problem of optimizing an equivalence class is studied and left and right sides of a dependency are shown to behave differently. A new representation for equivalence classes based on this observation is suggested. The optimization of single dependencies is shown to be NP-complete, and a method to produce good approximations is given.