The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity. We study a multiple commodity generalization of Max-Flow in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in \R^k. Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the mutual capacity of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with m constraints in at most \R^{(m-n+1)k} space from one that requires mn constraints in \R^{mk} space for a network with n nodes and m edges. We present efficient algorithms that compute eps-approximations to both the ratio and the integer ratio maximum flow problems.

翻译：最大流最小割定理是经典的最大流问题的对偶结果，该问题考虑单一商品的流量。我们研究最大流的多商品泛化形式，其中流量由实值k维向量构成，网络弧的容量由\R^k中的区域形成。鉴于多商品情形下缺乏明确的序关系定义，我们将广义最大流定义为所有流量值的可行域。我们定义了多商品环境下流与割的一系列概念与运算。研究一组割的互容量，即能通过该集合中所有割的流量集合。我们提出计算割对互容量的方法，并将其推广至任意割集合的计算方法。证明互容量恰好是网络中可行流的集合，因而等于最大流。此外，我们提供了一类简单的多商品最大流问题，其中利用此紧对偶结果的计算速度显著优于默认的暴力计算。我们还研究了更易处理的多商品最大流特例，其目标是通过网络输送给定向量的最大实数倍或整数倍。我们设计了一种增广圈搜索算法，将优化问题从需要包含n个节点和m条边的网络中\R^{mk}空间中的mn个约束，简化为至多\R^{(m-n+1)k}空间中的m个约束。我们提出高效算法，分别计算比值最大流问题和整数比值最大流问题的ε近似解。