Social Aware Coalition Formation with Bounded Coalition Size

In many situations when people are assigned to coalitions the assignment must be social aware, i.e, the utility of each person is the number of friends in her coalition. Additionally, in many situations the size of each coalition should be bounded. This paper initiates the study of such coalition formation scenarios. We show that finding a partition that maximizes the utilitarian social welfare is computationally hard, and provide a polynomial-time approximation algorithm. We also investigate the existence and the complexity of finding stable partitions. Namely, we show that there always exists a Nash Stable (NS) partition and the Contractual Strict Core (CSC) is never empty, but the Strict Core (SC) of some games is empty. Finding partitions that are NS or in the CSC is computationally easy, but finding partitions that are in the SC is hard. The analysis of the core is more involved. When the coalition size is bounded by 3 the core is never empty, and we present a polynomial time algorithm for finding a member of the core. In all other cases, we provide additive and multiplicative approximations of the core. In addition, we show in simulation over 100 million games that a simple heuristic always finds a partition that is in the core.