论文标题
均衡转运问题
Balanced flows for transshipment problems
论文作者
论文摘要
转运问题（g，d，λ）由有向图G =（v，e）建模，其加权顶点d =（d_v | v \ in v）和定向边缘λ=（λ_e| e \ in e）被解释为如下：g是：g是通信或运输网络，例如，e。 E中的每个边缘E \都是单向通信线，λ_e的道路或管道，而V中的每个顶点v \是生产节点d_v> 0，消耗d_v <0或过渡d_v = 0。非阴性流量x =（x_e \ nimide e \ in e n dion for dion for for dion for for for for for for for for for for for for for for for for for for for for for for for for for for for for for v for for v fl v fly v for v fl v fly for v f inf v \ y v \ n v \ in v \ in v \ n v \ in v \事件向V，等于d_v;或较短，如果A_G X = D，则A_G是G的顶点 - 边缘的发生率。但是，这种多余的被视为不需要的（危险），因此我们将多余比率向量r =（r_e = x_e = x_e /λ_e| e \ in e）上的词典作词上。更确切地说，首先，我们寻找弱可行的流量最小化e的最大值。在所有这样的流中，我们会寻找那些最大程度地减少R的坐标等等。显然，| e |这样的步骤定义了独特的平衡流，该流量为问题提供了Lexmin解决方案（g，d，λ）。我们在多项式时间内构造它，提供了向量d和λare整数。对于对称的挖掘，该问题由Gurvich和Gvishiani在1984年解决。在这里，我们将此结果扩展到了定向图。此外，我们简化了算法和证明，应用了1950年代后期由Gale和Hoffman获得的（G，D，λ）存在的经典标准。
A transshipment problem (G, d, λ) is modeled by a directed graph G = (V, E) with weighted vertices d = (d_v | v \in V) and directed edges λ= (λ_e | e \in E) interpreted as follows: G is a communication or transportation network, e.g., a pipeline; each edge e \in E is a one-way communication line, road or pipe of capacity λ_e, while every vertex v \in V is a node of production d_v > 0, consumption d_v < 0, or transition d_v = 0. A non-negative flow x = (x_e \mid e \in E) is called weakly feasible if for each v \in V the algebraic sum of flows, over all directed edges incident to v, equals d_v; or shorter, if A_G x = d, where A_G is the vertex-edge incidence matrix of G. A weakly feasible flow x is called feasible if x_e \leq λ_e for all e \in E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities x_e > λ_e are allowed. However, such an excess is viewed as unwanted (dangerous) and so we minimize the excess ratio vector r = (r_e = x_e / λ_e | e \in E) lexicographically. More precisely, first, we look for the weakly feasible flows minimizing the maximum of re over all e in E; among all such flows we look for those that minimize the second largest coordinate of r, etc. Clearly, |E| such steps define a unique balanced flow, which provides the lexmin solution for problem (G, d, λ). We construct it in polynomial time, provided vectors d and λare integer. For symmetric digraphs the problem was solved by Gurvich and Gvishiani in 1984. Here we extend this result to directed graphs. Furthermore, we simplify the algorithm and proofs applying the classic criterion of existence of a feasible flow for (G, d, λ) obtained by Gale and Hoffman in late 1950-s.
