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Unique column combinations of a relational database table are sets of columns that contain only unique values. Discovering such combinations is a fundamental research problem and has many different data management and knowledge discovery applications. Existing discovery algorithms are either brute force or have a high memory load and can thus be applied only to small datasets or samples. In this paper, the wellknown GORDIAN algorithm and "Apriori-based" algorithms are compared and analyzed for further optimization. We greatly improve the Apriori algorithms through efficient candidate generation and statistics-based pruning methods. A hybrid solution HCAGORDIAN combines the advantages of GORDIAN and our new algorithm HCA, and it significantly outperforms all previous work in many situations.
Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs). We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs. We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.