Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting

Inclusion/exclusion branching is a way to branch on requirements imposed on problems, in contrast to the classical branching on parts of the solution. The technique turned out to be useful for finding and counting (minimum) dominating sets (van Rooij et al., ESA 2009). In this paper, we extend the technique to the setting where one is given a set of properties and seeks (or wants to count) solutions that have at least a given number of these properties. Using this extension, we obtain the fastest exact algorithms for Partial Dominating Set and the parameterised problem k-Set Splitting. In particular, we apply the new idea to the fastest polynomial space algorithm for counting dominating sets, and directly obtain a polynomial space algorithm for Partial Dominating Set with the same running time up to a linear factor. Using the new approach combined with previous work, we also give a polynomial space algorithm for Set Splitting that improves the fastest known result significantly.