匹配与拟阵问题的代数结构与算法（初版）
Algebraic Structures and Algorithms for Matching and Matroid Problems (Preliminary Version)
摘要：
由Cunningham和Geelen（FOCS 1996）引入的基本路径匹配是拟阵交和非二部匹配的一种常见推广。本文的主要结果是基本路径匹配问题的一个新的代数刻画和一个在O（n^w）时间内构造基本路径匹配的算法，其中n是顶点数，w是矩阵乘法的指数。我们的算法是随机的，我们的方法假设给定的拟阵是线性的，可以表示在同一领域。我们的主要结果有几个特殊情况下的路径匹配问题的有趣的后果。对于拟阵的交，我们得到了一个运行时间为O（nr^（w-1））=O（nr ^1.38）的算法，其中拟阵有n个元素，秩为r.这改进了Gabow和Xu（FOCS 1989）提出的O（nr^1.62）的长期边界。此外，我们得到了一个简单的，纯代数算法的非二部匹配的运行时间为O（n^w）。这解决了Mucha和Sankowski（FOCS 2004）的核心开放问题。
英文摘要：
Basic path-matchings, introduced by Cunningham and Geelen (FOCS 1996), are a common generalization of matroid intersection and non-bipartite matching. The main results of this paper are a new algebraic characterization of basic path-matching problems and an algorithm for constructing basic path-matchings in O(n^w) time, where n is the number of vertices and w is the exponent for matrix multiplication. Our algorithms are randomized, and our approach assumes that the given matroids are linear and can be represented over the same field. Our main results have interesting consequences for several special cases of path-matching problems. For matroid intersection, we obtain an algorithm with running time O(nr^(w-1))=O(nr^1.38), where the matroids have n elements and rank r. This improves the long-standing bound of O(nr^1.62) due to Gabow and Xu (FOCS 1989). Also, we obtain a simple, purely algebraic algorithm for non-bipartite matching with running time O(n^w). This resolves the central open problem of Mucha and Sankowski (FOCS 2004).
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。
