有向图的入树覆盖
Covering Directed Graphs by In-trees
摘要：
给定一个有向图$D=(V,A)$，其中$d$指定顶点集$S=\{s_1,...,s_d\}\subseteq V$和函数$f\colon S \to \mathbb{Z}_+$，其中$\mathbb{Z}_+$表示非负整数集，我们考虑是否对每个$i=1,...,d$都存在用$T_{i,1},T_{i,2},..., T_{i,f(s_i)}$表示的$\sum_{i=1}^d f(s_i)$入树，使得$T_{i,1},...,T_{i,f(s_i)}$以$s_i$为根，每个$T_{i,j}$跨越从其可到达$s_i$的顶点，并且用于$i=1,...,d$和$j=1,...,f(s_i)$的$T_{i,j}$的所有弧集的并集覆盖$A$。在本文中，我们证明了这样的一组内树覆盖$A$可以找到通过使用一个算法的加权拟阵交问题的时间有界的一个多项式在$\sum_{i=1}^df(s_i)$和$D$的大小。此外，对于$D$是无环的情况下，我们提出了另一个特征的存在性的入树覆盖$A$，然后我们证明了入树覆盖$A$可以计算更有效地比一般情况下，通过寻找一系列的二分图的最大匹配。
英文摘要：
Given a directed graph $D=(V,A)$ with a set of $d$ specified vertices $S=\{s_1,...,s_d\}\subseteq V$ and a function $f\colon S \to \mathbb{Z}_+$ where $\mathbb{Z}_+$ denotes the set of non-negative integers, we consider the problem which asks whether there exist $\sum_{i=1}^d f(s_i)$ in-trees denoted by $T_{i,1},T_{i,2},..., T_{i,f(s_i)}$ for every $i=1,...,d$ such that $T_{i,1},...,T_{i,f(s_i)}$ are rooted at $s_i$, each $T_{i,j}$ spans vertices from which $s_i$ is reachable and the union of all arc sets of $T_{i,j}$ for $i=1,...,d$ and $j=1,...,f(s_i)$ covers $A$. In this paper, we prove that such set of in-trees covering $A$ can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in $\sum_{i=1}^df(s_i)$ and the size of $D$. Furthermore, for the case where $D$ is acyclic, we present another characterization of the existence of in-trees covering $A$, and then we prove that in-trees covering $A$ can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.
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