We thoroughly study a generalized version of the classic Stable Marriage and Stable Roommates problems where agents may share partners. We consider two prominent stability concepts: ordinal stability [Aharoni and Fleiner, Journal of Combinatorial Theory, 2003] and cardinal stability [Caragiannis et al., ACM EC 2019] and two optimality criteria: maximizing social welfare (i.e., the overall satisfaction of the agents) and maximizing the number of fully matched agents (i.e., agents whose shares sum up to one). After having observed that ordinal stability always exists and implies cardinal stability, and that the set of ordinally stable matchings in a restricted case admits a lattice structure, we obtain a complete picture regarding the computational complexity of finding an optimal ordinally stable or cardinally stable matching. In the process we answer an open question raised by Caragiannis et al. [AIJ 2020].

中文翻译：

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首选项下的分数匹配：稳定性和最优性

我们彻底研究了经典的“稳定婚姻和稳定室友”问题的广义版本，代理商可以在其中共享伙伴。我们考虑两个突出的稳定性概念：序数稳定性（Aharoni和Fleiner，《组合理论杂志》，2003年）和基数稳定性[Caragiannis等人，ACM EC 2019]以及两个最优性标准：最大化社会福利（即对社会的总体满意度）。代理商），并最大限度地增加完全匹配的代理商（即份额总和为一的代理商）的数量。在观察到序数稳定性始终存在并暗示基数稳定性之后，并且在有限情况下的一组常态稳定匹配包含了晶格结构，我们获得了关于找到最佳的常态稳定或基态稳定匹配的计算复杂度的完整描述。在这个过程中，我们回答了Caragiannis等人提出的一个开放性问题。[AIJ 2020]。