享乐多样性博弈：超过两种颜色的复杂性图景
Hedonic Diversity Games: A Complexity Picture with More than Two Colors
AI总结：
本文研究颜色数不限的享乐多样性博弈，通过新算法与下界刻画 Nash 稳定及个体稳定结果的完整参数化复杂性，并解决两颜色情形的开放问题。
摘要：
享乐多样性博弈是经典享乐博弈的一种变体，旨在更好地刻画涉及多样性与公平性的各类问题。以往工作主要针对两个多样性类别（在模型中表示为颜色）的情形，并给出了关于 Nash 稳定结果和个体稳定结果的一些初步复杂性理论与存在性结果。在本文中，我们设计了新的算法，并辅以相应的下界，从而围绕该问题最自然的参数化方式，给出了计算 Nash 稳定结果和个体稳定结果的完整参数化复杂性图景。关键在于，我们的结果适用于一般的享乐多样性博弈，其中颜色数量并不必然限于两种；结果表明，除两种平凡情形外，在这一设定下可处理性的必要条件是颜色数量受参数约束。此外，对于两种颜色的特殊情形，我们解决了以往工作（Boehmer and Elkind，AAAI 2020）提出的一个开放问题。
英文摘要：
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).
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