The stable marriage problem requires one to find a marriage with no blocking pair. Given a matching that is not stable, Roth and Vande Vate have shown that there exists a sequence of matchings that leads to a stable matching in which each successive matching is obtained by satisfying a blocking pair. The sequence produced by Roth and Vande Vate's algorithm is of length $O(n^3)$ where $n$ is the number of men (and women). In this paper, we present an algorithm that achieves stability in a sequence of matchings of length $O(n^2)$. We also give an efficient algorithm to find the stable matching closest to the given initial matching under an appropriate distance function between matchings.

中文翻译：

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稳定婚姻问题的改进稳定路径

稳定婚姻问题需要找到一个没有阻塞对的婚姻。给定一个不稳定的匹配，Roth 和 Vande Vate 已经证明存在导致稳定匹配的匹配序列，其中每个连续匹配都是通过满足分块对来获得的。Roth 和 Vande Vate 算法生成的序列长度为 $O(n^3)$，其中 $n$ 是男性（和女性）的数量。在本文中，我们提出了一种在长度为 $O(n^2)$ 的匹配序列中实现稳定性的算法。我们还给出了一种有效的算法，可以在匹配之间的适当距离函数下找到最接近给定初始匹配的稳定匹配。