Listing closed sets of strongly accessible set systems with applications to data

We study the problem of listing all closed sets of a closure operator @s that is a partial function on the power set of some finite ground set E, i.e., @s:F->F with [email protected]?P(E). 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 O(|E|(T"F+T"@s+|E|)) and space O(|E|+S"F+S"@s), where T"F, S"F, T"@s, and S"@s are the time and space complexities of checking membership in F and computing @s, 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.