关键词: Networks and graphs, maximum capacity, spanning arborescence,
polynomial algorithm

Abstract:

This article considers a class of bottleneck capacity expansion problems.
Such problems aim to enhance bottleneck capacity to a certain level with minimum cost. Given a network $G(V,A,\overline{C})$ consisting of
a set of nodes V={v₁,v₂,... ,v_n}, a set of arcs $A \subseteq \{(v_i,v_j)\ |\ i=1,2,\cdots ,n; ~j=1,2,\cdots ,n \}$ and a capacity vector $\overline{C}$. The component $\overline{c}_{ij}$ of $\overline{C}$ is the capacity of arc $(v_i,v_j)$. Define the capacity of a subset $A'$ of $A$ as the minimum capacity of the arcs in $A$, the capacity of a family ${\cal F}$ of subsets of $A$ is the maximum capacity of its members. There are two types of expanding models. In the arc-expanding model, the unit cost to increase the capacity of arc $(v_i,v_j)$ is $w_{ij}$. In the node-expanding model, it is
assumed that the capacities of all arcs $(v_i,v_j)$ which start at the same node v_i should be increased by the same amount and that the unit cost to make such expansion is w_i. This article considers three kinds of bottleneck capacity expansion problems (path, spanning arborescence and maximum flow) in both expanding models. For each kind of expansion problems,
this article discusses the characteristics of the problems and presents several results on the complexity of the problems.

Key words: Networks and graphs, maximum capacity, spanning arborescence,
polynomial algorithm