房屋分配问题中流行匹配的计数
Counting Popular Matchings in House Allocation Problems
摘要：
我们研究的问题，在一个给定的情况下，流行的匹配计数。一个流行的匹配实例由代理A和房屋H组成，其中每个代理根据他们的偏好对房屋的子集进行排名。匹配是指将代理人分配到房屋。匹配M比匹配M'更受欢迎，如果更喜欢M而不是M'的代理的数量多于更喜欢M'而不是M的人数。如果不存在比M更流行的匹配，则匹配M被称为流行的。McDermid和Irving给出了一个多时间算法，用于在偏好列表严格排序时计算流行匹配的数量。我们首先考虑偏好列表中的关系。Nasre证明了当存在关系时，计算流行匹配的数量是#P-hard问题。我们对此问题给出了一个FPRAS。然后，我们考虑流行的匹配问题，偏好列表是严格有序的，但每个房子有一个与之相关的容量。我们给出了一个开关图表征流行的匹配在这种情况下。这样的刻画在之前的严格有序偏好列表（McDermid和Irving）和带关系偏好列表（Nasre）的情况下进行了研究。我们使用我们的特性证明，计数流行的匹配能力的情况下是P-困难的。
英文摘要：
We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M' if the number of agents that prefer M to M' is more than the number of people that prefer M' to M. A matching M is called popular if there exists no matching more popular than M. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem. We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard.
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