最大偶因子问题的一个快速算法
A Faster Algorithm for the Maximum Even Factor Problem
摘要：
给定一个有向图$G = (VG,AG)$，一个\emph{even factor} $M \subseteq AG$是一个弧的子集，它可以分解成一个节点不相交的路径和偶圈的集合。Geleen和Cunningham在有向图中引入了偶因子，推广了无向图中的路匹配.在一般有向图中寻找最大基数的偶因子是NP-难的，但对于\emph{odd-cycle symmetric}类有向图，这个问题是多项式可解的。到目前为止，唯一已知的用于此任务的组合算法是由于Pap;它具有$O(n^4)$的运行时间（以下$n$代表$G$中的节点数）。在本文中，我们提出了一种新的\emph{sparse recovery}技术，并设计了一个$O(n^3 \log n)$ -时间算法，以寻找一个最大基数偶因子的奇循环对称有向图。
英文摘要：
Given a digraph $G = (VG,AG)$, an \emph{even factor} $M \subseteq AG$ is a subset of arcs that decomposes into a collection of node-disjoint paths and even cycles. Even factors in digraphs were introduced by Geleen and Cunningham and generalize path matchings in undirected graphs. Finding an even factor of maximum cardinality in a general digraph is known to be NP-hard but for the class of \emph{odd-cycle symmetric} digraphs the problem is polynomially solvable. So far, the only combinatorial algorithm known for this task is due to Pap; it has the running time of $O(n^4)$ (hereinafter $n$ stands for the number of nodes in $G$). In this paper we present a novel \emph{sparse recovery} technique and devise an $O(n^3 \log n)$-time algorithm for finding a maximum cardinality even factor in an odd-cycle symmetric digraph.
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