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).

翻译：享乐多样性博弈是经典享乐博弈的变种，旨在更准确地建模关于多样性和公平性的诸多问题。先前的研究主要针对具有两个多样性类别（模型中用颜色表示）的情形，并给出了关于纳什均衡和个体稳定结果的一些初步复杂性理论和存在性结论。在此，我们设计了新算法并辅以下界，针对问题最自然的参数化方式，为计算纳什均衡与个体稳定结果提供了完整的参数化复杂性图景。关键的是，我们的结果适用于颜色数量不一定限于两种的广义享乐多样性博弈，并表明——除两个平凡情形外——在该设定中可计算性的必要条件是颜色数量受参数限制。此外，对于两种颜色的特殊情况，我们解决了先前工作（Boehmer 与 Elkind，AAAI 2020）中提出的一个未决问题。