Hedonic Diversity Games: A Complexity Picture with More than Two Colors

TL;DR

This paper studies the computational complexity of finding stable outcomes in Hedonic diversity games with multiple colors, providing algorithms and bounds that extend previous two-color results to more general cases.

Contribution

It introduces a complete parameterized complexity analysis for Hedonic diversity games with more than two colors, including new algorithms and complexity bounds.

Findings

Algorithms with matching lower bounds for Nash and stable outcomes

Complexity results for general multi-color cases

Resolution of an open question for two-color scenarios

Abstract

Hedonic diversity games are a variant of the classical Hedonic games designed to better model a variety of questions concerning diversity and fairness. Previous works mainly targeted the case with two diversity classes (represented as colors in the model) and provided some initial complexity-theoretic and existential results concerning Nash and individually stable outcomes. Here, we design new algorithms accompanied with lower bounds which provide a complete parameterized-complexity picture for computing Nash and individually stable outcomes with respect to the most natural parameterizations of the problem. Crucially, our results hold for general Hedonic diversity games where the number of colors is not necessarily restricted to two, and show that -- apart from two trivial cases -- a necessary condition for tractability in this setting is that the number of colors is bounded by the parameter. Moreover, for the special case of two colors we resolve an open question asked in previous work (Boehmer and Elkind, AAAI 2020).