一种二分稳定匹配优化的新方法
A new approach to bipartite stable matching optimization
AI总结：
针对二分稳定匹配优化问题，提出基于强多项式网络流的算法计算ℓ个不相交且总成本最小的稳定匹配，通过将其表示为有向图割或偏序集反链，结合相关定理，还得到覆盖所有稳定边的极小-极大公式。
摘要：
作为先前已解决的二分稳定匹配相关优化问题的通用推广，我们提出一种基于强多项式网络流的算法，用于计算具有最小总成本的ℓ个不相交稳定匹配。该方法的主要观察是：稳定匹配作为边集，可表示为关联有向图的特定割。这使我们能直接利用不相交割的结果来回答不相交稳定匹配的问题。我们还构造了一种将稳定匹配表示为偏序集（poset）中最大规模反链的方法，从而可直接将Dilworth、Mirsky、Greene和Kleitman的定理应用于稳定匹配。这些方法的另一结果是得到覆盖所有稳定边所需最少稳定匹配数的极小-极大公式。
英文摘要：
As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing $\ell$ disjoint stable matchings with minimum total cost. The major observation behind the approach is that stable matchings, as edge sets, can be represented as certain cuts of an associated directed graph. This allows us to use results on disjoint cuts directly to answer questions about disjoint stable matchings. We also provide a construction that represents stable matchings as maximum-size antichains in a partially ordered set (poset), which enables us to apply the theorems of Dilworth, Mirsky, Greene and Kleitman directly to stable matchings. Another consequence of these approaches is a min-max formula for the minimum number of stable matchings covering all stable edges.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
