On Enumerating Frequent Closed Patterns with Key in Multi-relational Data

We study the problem of mining closed patterns in multi-relational databases. Garriga et al. (IJCAI'07) proposed an algorithm RelLCM2 for mining closed patterns (i.e., conjunctions of literals) in multi-relational data, which is an extension of LCM, an efficient enumeration algorithm for frequent closed itemsets mining proposed in the seminal paper by Uno et al. (DS'04). We assume that a database considered contains a special predicate called key (or target), which determines the entities of interest and what is to be counted. We introduce a notion of closed patterns with key (key-closedness for short), where variables in a pattern other than the one in a key predicate are considered to be existentially quantified, and they are linked to a given target object. We then define a closure operation (key-closure) for computing key-closed patterns, and show that the difference between the semantics of key-closed patterns and that of the closed patterns in RelLCM2 implies different properties of the closure operations; in particular, the uniqueness of closure does not hold for key-closure. Nevertheless, we show that we can enumerate key-closed patterns using the technique of ppc-extensions a la LCM, thereby making the enumeration possible without storage space for previously generated patterns. We also propose a literal order designed for mining key-closed patterns, which will require less search space. The correctness of our algorithm is shown, and its computational complexity is discussed. Some preliminary experimental results are also given.