论文标题
Birkhoff-von Neumann定理扩展到非双向图
An Extension of the Birkhoff-von Neumann Theorem to Non-Bipartite Graphs
论文作者
论文摘要
我们证明，可以在多项式时间内写入非两分图中的分数完美匹配，作为完美匹配的凸组合。这将Birkhoff-Von Neumann定理从两部分扩展到非双分化图。 Birkhoff和Von Neumann的算法是贪婪的。它从给定的分数完美匹配开始，然后从中依次“删除”完美的匹配，并具有适当的系数。这是在非双方图中失败的 - 任意删除完美的匹配可能会导致一个非空的图形，但没有完美的匹配。使用奇数剪裁适当地节省了一天。
We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively "removes" from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriately saves the day.
