Let G = (V, E) be a flow network
with source s, sink t, and capacity function c(·). Assume that, for every
edge e ∈ E, c(e) is an integer. Define the size of an s-t cut (A, B) in G
to be the number of edges directed from A to B. Our goal is to identify,
from among all minimum cuts in G, a minimum cut whose size is as small
as possible.
Let us define a new capacity function c'(·) for G as follows. For each
edge e ∈ E, by c'(e) = m·c(e)+1. Suppose (A, B) is a minimum
cut in in G with respect to the capacity function c'(·).
(a) Show that (A, B) is a minimum cut with respect to the original capacity
function c(·).
(b) Show that, amongst all minimum cuts in G, (A, B) is a cut of smallest
size.
(c) Use the results of parts (a) and (b) to obtain a polynomial-time algorithm
to find a minimum cut of smallest size in a flow network.
如何为此编写多项式时间算法？任何的想法？
我不会破坏答案，但我会留意任何未来发现这篇文章的学生。考虑一下如果在G中采用两个最小割（A，B）和（C，D）会发生什么，这样一个边的边数最小，而另一边的边数不是。然后将它们映射到G'并考虑这两个切割的值。
搜索dijkstra的算法，它通常用于图中的最短路径。我不完全理解你想要实现的算法，但我觉得它非常相似，可以使用dijstra背后的思考