Hedonic games are an archetypal problem in coalition formation, where a set of selfish agents want to partition themselves into stable coalitions. In this work, we focus on two natural constraints on the possible outcomes. First, we require that exactly k coalitions are created. Then, loosely following the model of Bilò et al. (AAAI 2022), we assume that each of the k coalitions is additionally associated with a lower and upper bound on its size. The notion of stability that we study is that of individual rationality (IR), which requires that no agent strictly prefers to be alone compared to being in his or her coalition. Although IR is trivially satisfiable even in the most general models of hedonic games, the complexity picture of deciding whether an IR allocation exists, considering the above constraints, is unexpectedly rich. We reveal that tractable fragments of this computational problem require surprisingly nontrivial arguments, even if we restrict ourselves to additively separable and fractional hedonic games. Our tractability results, achieved by exploiting the structure of the underlying preference graph, are also complemented by their intractability counterparts, painting a fairly complete picture of the tractability landscape of this problem.

翻译：享乐博弈是联盟形成中的典型问题，其中一组自私的参与者希望将自己划分为稳定的联盟。本文重点研究了可能结果上的两种自然约束。首先，我们要求恰好形成k个联盟。其次，大致遵循Bilò等人（AAAI 2022）的模型，我们假设每个k个联盟还额外关联一个规模的下界和上界。我们研究的稳定性概念是个体理性（IR），即要求没有参与者严格偏好独自一人而非留在其所在联盟。尽管个体理性在最一般的享乐博弈模型中平凡地可满足，但在考虑上述约束时，判定是否存在IR分配的计算复杂性图谱却意外丰富。我们发现，即使局限于可加可分离和分数享乐博弈，这一计算问题的易处理片段也需要相当不平凡的论证。通过利用底层偏好图的结构，我们取得了易处理性结果，同时结合相应的难处理性结果，为该问题的易处理性图景提供了相当完整的刻画。