Parameterized Complexity of Group Activity Selection.

We consider the Group Activity Selection Problem (GASP) in which a group of agents need to be assigned to activities, subject to agent preferences and stability conditions. In GASP, the agents announce dichotomic preferences on which (activity, number-of-participant) pairs are acceptable to them. We consider five solution concepts of assignments: (1) individual rationality (everyone who is assigned to an activity is willing to participate), (2) (Nash) stability (no agent wants to deviate from the assignment), (3) envy-freeness (no agent is envious of someone else's assignment), (4) stability and envy-freeness, and (5) perfection (everyone is assigned and willing to participate). It is known that finding an assignment of a given size with any of these properties is NP-complete. We study the complexity of GASP on a finer scale, through the lens of parameterized complexity. We show that the solution concepts above differ substantially, when parameterized by the size of the solution (the number of assigned agents or the number of used activities). In particular, finding an individually rational assignment is fixed parameter tractable, yet other solutions concepts are less tractable (W[1]- and W[2]-hard) even under very natural restrictions on inputs.