Group Activity Selection with Few Agent Types

Abstract

In this paper we establish the complexity map for the Group Activity Selection Problem (GASP), along with two of its prominent variants called sGASP and gGASP, focusing on the case when the number of types of agents is the parameter. In all these problems, one is given a set of agents (each with their own preferences) and a set of activities, and the aim is to assign agents to activities in a way which satisfies certain global as well as preference-based conditions. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps that allow us to focus on solutions with a simpler, well-defined structure (in particular, they are “acyclic”). These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta et al. (in: Algorithmic game theory—10th international symposium, SAGT 2017, vol 10504 of lecture notes in computer science, Springer, 2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well.