Group Activity Selection on Social Networks

We propose a new variant of the group activity selection problem (GASP), where the agents are placed on a social network and activities can only be assigned to connected subgroups (gGASP). We show that if multiple groups can simultaneously engage in the same activity, finding a stable outcome is easy as long as the network is acyclic. In contrast, if each activity can be assigned to a single group only, finding stable outcomes becomes computationally intractable, even if the underlying network is very simple: the problem of determining whether a given instance of a gGASP admits a Nash stable outcome turns out to be NP-hard when the social network is a path or a star, or if the size of each connected component is bounded by a constant. We then 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 standard 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 standard 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.
Comment: Preliminary versions of this article appeared in Proceedings of the 31st AAAI Conference on Artificial Intelligence (Igarashi et al., 2017), and Proceedings of the 16th International Joint Conference on Autonomous Agents and Multi Agent Systems (Igarashi et al., 2017). In this longer version, we provide all proofs that were omitted from the conference version and improve some of the results