How Hard Is It to Control A Group?

TL;DR

This paper investigates the computational complexity of controlling group qualification processes under various rules, analyzing how societal changes affect outcomes and providing complexity classifications including polynomial-time and NP-hard results.

Contribution

It offers a comprehensive complexity analysis of control problems in group identification models, including polynomial-time, NP-hard, and fixed-parameter tractability results.

Findings

Polynomial-time solvability for some control problems.

NP-hardness results for other control scenarios.

Fixed-parameter tractability in certain cases.

Abstract

We consider group identification models in which the aggregation of individual opinions concerning who is qualified in a given society determines the set of socially qualified persons. In this setting, we study the extent to which social qualification can be changed when societies expand, shrink, or partition themselves. The answers we provide are with respect to the computational complexity of the corresponding control problems and fully cover the class of consent aggregation rules introduced by Samet & Schmeidler (2003) as well as procedural rules for group identification. We obtain both polynomial-time solvability results and NP-hardness results. In addition, we also study these problems from the parameterized complexity perspective, and obtain some fixed-parameter tractability results.