覆盖约束下的稳定婚姻：一个完全的计算三分法
Stable Marriage with Covering Constraints: A Complete Computational Trichotomy
摘要：
我们考虑具有覆盖约束的稳定婚姻（SMC）：在稳定婚姻的这种变体中，我们区分了一个女性子集和一个男性子集，并且我们寻求匹配所有杰出人物的最少阻塞对的匹配。我们研究如何一组自然参数，即最大长度的偏好列表的男性和女性，杰出的男性和女性的数量，以及允许的阻塞对的数量确定这个问题的计算易处理性。我们的主要结果是一个完整的复杂性剖分，对于所研究的参数的每一个选择，分类SMC多项式时间可解，NP-硬和固定参数易处理，或NP-硬和W[1]-硬。我们还对所有可能仅区分女性的片面约束情况进行了分类。
英文摘要：
We consider Stable Marriage with Covering Constraints (SMC): in this variant of Stable Marriage, we distinguish a subset of women as well as a subset of men, and we seek a matching with fewest number of blocking pairs that matches all of the distinguished people. We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy that, for each choice of the studied parameters, classifies SMC as polynomial-time solvable, NP-hard and fixed-parameter tractable, or NP-hard and W[1]-hard. We also classify all cases of one-sided constraints where only women may be distinguished.
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