We show that given a SM instance G as input we can find a largest collection of pairwise edge-disjoint stable matchings of G in time linear in the input size. This extends two classical results: 1. The Gale-Shapley algorithm, which can find at most two ("extreme") pairwise edge-disjoint stable matchings of G in linear time, and 2. The polynomial-time algorithm for finding a largest collection of pairwise edge-disjoint perfect matchings (without the stability requirement) in a bipartite graph, obtained by combining K\"{o}nig's characterization with Tutte's f-factor algorithm. Moreover, we also give an algorithm to enumerate all maximum-length chains of disjoint stable matchings in the lattice of stable matchings of a given instance. This algorithm takes time polynomial in the input size for enumerating each chain. We also derive the expected number of such chains in a random instance of Stable Matching.

翻译：我们显示,作为输入的 SM 实例 G, 我们可以在输入大小中找到最大的对齐边缘不连接稳定匹配的集合。 这延伸了两个经典结果 : 1. Gale- Shapley 算法, 最多可以在两个( “ Exreme ” ) 中找到对齐边缘不连接稳定匹配的对齐边缘不连接稳定匹配的匹配, 以及 2. 在双边图形中找到一个最大的对齐边缘不连接匹配的集合的多元时间算法( 没有稳定要求 ), 这是通过将 K\ “ { o} nig ” 的特性与 Tutte f- factor 算法结合而获得的 。 此外, 我们还给出一种算法, 将所有不连接稳定匹配的最长长度匹配链在某个特定实例的边框中进行计算 。 这个算法在输入大小中需要时间的多数值, 来计算每个链。 我们还在 Stable 匹配的随机实例中得出这些链的预期数目 。