We study the problem of listing all closed sets of a closure operator that is a partial function on the power set of some finite ground set , i.e., with . A very simple divide-and-conquer algorithm is analyzed that correctly solves this problem if and only if the domain of the closure operator is a strongly accessible set system. Strong accessibility is a strict relaxation of greedoids as well as of independence systems. This algorithm turns out to have delay and space , where , , , and are the time and space complexities of checking membership in and computing , respectively. In contrast, we show that the problem becomes intractable for accessible set systems. We relate our results to the data mining problem of listing all support-closed patterns of a dataset and show that there is a corresponding closure operator for all datasets if and only if the set system satisfies a certain confluence property.
An early version of this paper appeared in the Proc. of the 11th European Conference on Principles and Practice of Knowledge Discovery in Databases (PKDD), in: LNAI, vol. 4702, Springer-Verlag, Heidelberg, 2007, pp. 382–389.