Group Activity Selection on Social Networks

TL;DR

This paper studies the computational complexity of a social network-based group activity selection problem, revealing tractable cases on acyclic networks and hardness results on simple network structures.

Contribution

It introduces a new variant of GASP considering social network constraints and analyzes its complexity, providing algorithms and hardness results for various stability notions.

Findings

Finding stable outcomes is easy on acyclic networks with multiple activities.

Determining Nash stability is NP-hard on paths and stars with single activities.

An FPT algorithm is proposed for Nash stability when the network is acyclic and the number of activities is small.

Abstract

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.