AI 中文总结
该研究探讨朋友、敌人与中立者构成的受限享乐博弈中，满足个体理性及规模约束的联盟划分问题，分析以朋友/敌人为导向偏好下的计算复杂度，明确了不同关系结构对问题可处理性的影响。
AI 中文摘要
我们研究由朋友、敌人和中立者构成的博弈中的受限联盟形成问题，采用可加可分偏好的两种标准细化类型：以朋友为导向和以敌人为导向。我们要求划分满足个体理性（IR），同时额外要求恰好存在k个非空联盟，每个联盟的规模满足规定的上下限。尽管对于任何享乐博弈，仅满足IR条件是极易实现的，但规模约束使得判定是否存在可行划分的问题在计算上变得难以处理。两种模型呈现出截然不同的情况：在以敌人为导向的偏好下，该问题可归约为规模受限的图着色问题，其复杂度也随之确定；然而在以朋友为导向的偏好下，情况远为复杂，其复杂度由敌对关系结构而非友谊结构决定，且进一步受两个因素影响：施加的规模要求的严格程度，以及关系是对称还是非对称，其中若干情况在对称设定下可处理，但一旦允许非对称则变得难以处理。我们从经典复杂度和参数化复杂度两个维度绘制该边界，完整阐明了朋友/敌人结构的哪些属性导致了问题的难解性。
英文摘要
We study constrained coalition formation in games induced by friends, enemies, and neutrals, under the two standard refinements of additively separable preferences: friend-oriented and enemy-oriented. We ask for partitions that are individually rational (IR), while additionally requiring exactly $k$ non-empty coalitions, each satisfying a prescribed lower and upper bound on its size. Although IR alone is trivial to satisfy for any hedonic game, the size constraints make it computationally intractable to decide whether a feasible partition exists. The two models tell strikingly different stories. Under enemy-oriented preferences, the problem collapses to size-constrained graph coloring, and its complexity follows accordingly. Under friend-oriented preferences, however, the picture is far more intricate, and is governed by the enmity structure rather than the friendships. The complexity is further shaped by two factors: how strict the imposed size requirements are, and whether relationships are symmetric or asymmetric, with several cases turning out tractable in the symmetric setting but intractable once asymmetry is allowed. Charting this boundary in terms of both classical and parameterized complexity, we provide a complete understanding of which properties of the friend/enemy structure are responsible for hardness.
