无向网络中的最小费用多流问题
On Min-Cost Multiflow Problem in Node-Capacitated Undirected Networks
摘要：
我们考虑一个无向图$G = (VG, EG)$，它有一个终端集$T \subseteq VG$，节点的容量$c(v)$和代价$a(v)$为非负整数$v\in VG$ .如果$G$中的路径的端点是不同的终端，则该路径是\emph{$T$-path}。通过\emph{multiflow}，我们表示函数$F$，它为每个$T$ -路径$P$分配非负有理\emph{weight} $F(P)$，如果通过每个节点$v$的$T$ -路径的权重之和不超过$c(v)$，则多流被称为\emph{feasible}. $F$的\emph{value}是权重$F(P)$的和，并且$F$的\emph{cost}是$F(P)$乘以$P$相对于r.t. $a$，在所有$T$ -路径$P$上。推广已知的结果，边容量限制的多流，我们发现的问题，找到一个最小的成本多流之间的可行的多流的最大可能值承认\emph{half-integer}最佳原始和对偶解决方案。此外，我们设计了一个强多项式算法来寻找这样的最优解。
英文摘要：
We consider an undirected graph $G = (VG, EG)$ with a set $T \subseteq VG$ of terminals, and with nonnegative integer capacities $c(v)$ and costs $a(v)$ of nodes $v\in VG$. A path in $G$ is a \emph{$T$-path} if its ends are distinct terminals. By a \emph{multiflow} we mean a function $F$ assigning to each $T$-path $P$ a nonnegative rational \emph{weight} $F(P)$, and a multiflow is called \emph{feasible} if the sum of weights of $T$-paths through each node $v$ does not exceed $c(v)$. The \emph{value} of $F$ is the sum of weights $F(P)$, and the \emph{cost} of $F$ is the sum of $F(P)$ times the cost of $P$ w.r.t. $a$, over all $T$-paths $P$. Generalizing known results on edge-capacitated multiflows, we show that the problem of finding a minimum cost multiflow among the feasible multiflows of maximum possible value admits \emph{half-integer} optimal primal and dual solutions. Moreover, we devise a strongly polynomial algorithm for finding such optimal solutions.
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