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3. a.
Prove that for all maximum flows f, there exists a minimum capacity cut M such that for all
edges e, f(e) = c[e]
Let’s consider the paths from on a graph with some maximum flow from s to t and consider the
minimum capacity edge e = (u,v) on one such path.
Lemma 1: The set of edges with saturated flow satisfy the above property that they are the set of
minimum capacity edges of their paths.
If there is an e’ on the path containing an edge from the above set that has a lower capacity than
our edge, then either/or
1.
e' came before our edge in consideration (saturated flow), in which case the total capacity
must have been less than that of e, which means that our edge couldn’t have been
saturated.
2.
e' came after our edge in consideration, in which case the total inbound flow to e’ must
have been more than what it could handle.
Both cases are contradictions; hence the set of edges with saturated flow M must satisfy the
property that they are the set of minimum capacity edges of the paths. Now if we partition G into
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 Spring '08
 KLEINBERG
 Algorithms

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