词典偏好下多伙伴匹配问题的强核心解和帕累托最优解
Strong core and Pareto-optimal solutions for the multiple partners matching problem under lexicographic preferences
摘要：
在多个合作伙伴匹配问题中，代理可以有多个合作伙伴。在本文中，我们考虑了双边多对多稳定匹配问题和单边稳定夹具问题下的字典偏好。我们研究强核心和帕累托最优的解决方案，从计算的角度来看，这种设置。首先，我们提供了一个例子来表明，强核可以是空的，即使在这些严格的限制下的多对多的问题，并决定非空的强核是NP-难的。我们还表明，对于一个给定的匹配检查帕累托最优性和强核心属性是多对多问题的合作NP完全问题，并决定一个完整的帕累托最优匹配的存在也是NP难的夹具问题。在积极的一面，我们给出了有效的算法，找到一个近可行的强核心的解决方案，其中的能力只违反了最多一个单位为每个代理，并找到一个半匹配的强核心的分数匹配。这些多项式时间算法是基于顶部交易周期算法。最后，我们还表明，找到一个最大的尺寸匹配，是帕累托最优的，可以有效地进行多对多的问题，这是在对比的硬度结果的夹具问题。
英文摘要：
In a multiple partners matching problem the agents can have multiple partners up to their capacities. In this paper we consider both the two-sided many-to-many stable matching problem and the one-sided stable fixtures problem under lexicographic preferences. We study strong core and Pareto-optimal solutions for this setting from a computational point of view. First we provide an example to show that the strong core can be empty even under these severe restrictions for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. We also show that for a given matching checking Pareto-optimality and the strong core properties are co-NP-complete problems for the many-to-many problem, and deciding the existence of a complete Pareto-optimal matching is also NP-hard for the fixtures problem. On the positive side, we give efficient algorithms for finding a near feasible strong core solution, where the capacities are only violated by at most one unit for each agent, and also for finding a half-matching in the strong core of fractional matchings. These polynomial time algorithms are based on the Top Trading Cycle algorithm. Finally, we also show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems, which is in contrast with the hardness result for the fixtures problem.
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