Comments on “Problem Decomposition Method to Compute an Optimal Cover for a Set of Functional Dependencies”

Maier, in his seminal paper [1], presents that the problem of finding an optimal cover is NP-complete. Cotelea’s paper [2] proposes a problem decomposition method for finding an optimal cover. The basic idea of the method is: (1) a big and intractable problem (optimal cover problem) is broken down into some smaller problems; (2) particular solutions of these smaller problems are combined to construct the initial problem solution. More specifically, the main idea of the method is: (1) a relational schema R with a set F of functional dependencies is partitioned in some subschema; (2) each subschema will be found determinants with the fewest attributes (including repeated); (3) substituting the groups of attributes in F that are determinants in some subschema with the shortest determinants, Cotelea states that this happens only for equivalence classes of functional dependencies containing the determinants in the left or right sides as subsets. The idea embodied in this method is classic in the algorithm theory area, and Cotelea proposes many valuable results. However, are these sufficient for finding all optimal covers? (Seemingly unlikely) Please see a counterexample as the following. Example 1. Given G={AD→E, A→BC, B→A, C→A} that is a minimum and reduced set of functional dependencies, please find an optimal cover for G. For example, H={AD→E, A→B, B→C, C→A} is an optimal cover for G. The contribution graph of G is shown in Figure 1.