Extending Set-Based Dualization: Application to Pattern Mining
Résumé
Dualization problems have been intensively studied in combinatorics, AI and pattern mining for years. Roughly speaking, for a partial order $(P,\preceq)$ and some monotonic predicate $Q$ over $P$, the dualization consists in identifying all maximal elements of $P$ verifying $Q$ from all minimal elements of $P$ not verifying $Q$, and vice versa.
The dualization is equivalent to the enumeration of minimal transversal of hypergraphs whenever $(P,\preceq)$ is a boolean lattice.
In the setting of interesting pattern mining in databases, $P$ represents a set of patterns and whenever $P$ is isomorphic to a boolean lattice, the pattern mining problem is said to be \emph{representable as sets}. The class of such problems is denoted by \RAS.
In this paper, we introduce a \emph{weak representation as sets} for pattern mining problems which extends the \RAS\ class to a wider and significantly larger class of problems, called \WRAS.
We also identify \EWRAS, an \emph{efficient} subclass of \WRAS\ for which the dualization problem is still quasi-polynomial.
Finally, we point out that one representative pattern mining problem known not to be in \RAS, namely \emph{frequent rigid sequences with wildcard}, belongs to \EWRAS.
These new classes might prove to have large impact in unifying existing pattern mining approaches.