On the Complexity of Maximal/Closed Frequent Tree Mining for Bounded Height Trees
Kenta Komoto, Kazuhiro Kurita, Hirotaka Ono
TL;DR
The paper investigates the complexity of enumerating frequent maximal and closed trees under bounded height, resolving gap-filled boundaries between tractable and intractable cases. It provides a polynomial-delay, polynomial-space algorithm for Closed Frequent Tree Mining in the unordered setting with height $2$, and establishes Dualization-based hardness that precludes general output-polynomial algorithms for several related problems, including Maximal Frequent Tree Mining in both ordered and unordered settings under tight height constraints. The results delineate a sharp contrast: tractability for certain closed and common-tree subproblems at very low heights, versus strong intractability (P vs NP) for maximal frequent tree enumeration as height grows or in the ordered setting. These findings clarify practical expectations for tree-mining tasks in domains like XML, and guide algorithm design toward height-bounded or specialized cases. Future work is invited to close the gap for height $3$ and beyond in Closed Frequent Tree Mining and related problems.
Abstract
In this paper, we address the problem of enumerating all frequent maximal/closed trees. This is a classical and central problem in data mining. Although many practical algorithms have been developed for this problem, its complexity under ``realistic assumptions'' on tree height has not been clarified. More specifically, while it was known that the mining problem becomes hard when the tree height is at least 60, the complexity for cases where the tree height is smaller has not yet been clarified. We resolve this gap by establishing results for these tree mining problems under several settings, including ordered and unordered trees, as well as maximal and closed variants.