On Parameterized Complexity of Group Activity Selection Problems on Social Networks.

In Group Activity Selection Problem with graph structure (gGASP), players form coalitions to participate in activities and have preferences over pairs of the form (activity, group size); moreover, a group of players can only engage in the same activity if the members of the group form a connected subset of the underlying communication structure. We study the parameterized complexity of finding outcomes of gGASP that are Nash stable, individually stable or core stable. For the parameter `number of activities', we propose an FPT algorithm for Nash stability for the case where the social network is acyclic and obtain a W[1]-hardness result for cliques (i.e., for classic GASP); similar results hold for individual stability. In contrast, finding a core stable outcome is hard even if the number of activities is bounded by a small constant, both for classic GASP and when the social network is a star. For the parameter `number of players', all problems we consider are in XP for arbitrary social networks; on the other hand, we prove W[1]-hardness results with respect to the parameter `number of players' for the case where the social network is a clique (i.e., for classic GASP).