On Problems With Closure Properties

We consider a specific class of combinatorial search or optimization problems where the search space gives rise to a closure operator and essentially the hulls are the only relevant subsets that must be checked in a brute force approach. We suggest that such a closure property can help to reduce time complexities. Moreover we propose two types of (structural) parameterization of instance classes based on the closure property and outline how it could be used to achieve FPT characterizations. In this setting, three example problems are considered: a covering problem from combinatorial geometry, a variant of the autarky problem in propositional logic, and a specific graph problem on finite forests.