Finding and recognizing popular coalition structures

An important aspect of multi-agent systems concerns the formation of coalitions that are stable or optimal in some well-defined way. The notion of popularity has recently received a lot of attention in this context. A partition is popular if there is no other partition in which more agents are better off than worse off. In 2019, a long-standing open problem concerning popularity was solved by proving that computing popular partitions in roommate games is NP-hard, even when preferences are strict. We show that this result breaks down when allowing for randomization: mixed popular partitions can be found efficiently via linear programming and a separation oracle. Mixed popular partitions are particularly attractive because they are guaranteed to exist in any coalition formation game. Our result implies that one can efficiently verify whether a given partition in a roommate game is popular and that strongly popular partitions can be found in polynomial time (resolving an open problem). By contrast, we prove that both problems become computationally intractable when moving from coalitions of size 2 to coalitions of size 3, even when preferences are strict and globally ranked. Moreover, we give elaborate proofs showing the NP-hardness of finding popular, strongly popular, and mixed popular partitions in additively separable hedonic games and finding popular partitions in fractional hedonic games.