Combinatorial Problems With Closure Structures

Abstract

We consider a specific class of combinatorial search respectively 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 structure can help to reduce time complexities. Moreover we propose two types of (structural) parameterizations of instance classes based on the closure property and outline how it could be used to achieve fixed-parameter tractability (FPT) characterizations. In this setting, three example problems are described: a covering problem from combinatorial geometry, a variant of the autarky problem in propositional logic, and a specific graph problem on finite forests.