The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for min bp hrc. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of min bp hrc. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered.

中文翻译：

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夫妻在医院/居民问题中的“几乎稳定”匹配

夫妇的医院/居民问题（hrc）对打算将初级医生分配到允许夫妇在成对的（通常在地理上接近的）医院上提交联合偏好列表的医院进行建模。已知不需要稳定的匹配，因此我们考虑min bp hrc，即找到允许最小数量的阻塞对（即“尽可能稳定”）的匹配的问题。我们证明，即使在每对夫妇都只能接受一对医院接受的高度严格的情况下，这个问题仍然很难解决，并且难以近似。但是，如果我们进一步假设每个居民和医院的偏好列表的长度最多为2，则针对这种情况，我们将给出多项式时间算法。然后，我们为min bp hrc提出第一个整数编程（IP）和约束编程（CP）模型。最后，我们讨论了适用于随机生成的min bp hrc实例的这些模型的经验评估。我们发现，平均而言，CP模型比IP模型快1.15倍，而对CP模型应用预求解时，平均速度要快8.14倍。我们进一步观察到，对于所考虑的（28,000）个实例，解决方案允许的阻塞对的数量非常小，即通常最多1个，并且永远不超过2个。