Math 482 (Lecture 17): Max flow and min cut II
This lecture discusses Sections 6.1 and 6.2 of the textbook.
Questions we had left open in lecture 16:
Q1. How do you find an augmenting path?
Q2. Prove that if there does not exist an augmenting path then the current flow is indeed
optimal.
Q3. Given an optimal flow, how do you find a min cut?
In class exercise/discussion:
(Q1) How do you find an augmenting path?
Brief discussion notes/main ideas:
1. Construct the residual graph G'.
2. Put s into the set LABEL of "labelled vertices". Label all vertices v such that (s,v) is an
arc in G' that is not saturated, add all such v to LABEL.
3. More generally, go through all vertices x in LABEL that haven't yet been inspected. A
vertex v not in LABEL is added if either there is an arc in G' from x to v that is
unsaturated: this means (x,v) is a backward arc in G' but not in the original G OR f(x,v)<
b(x,v).
4. TERMINATE when either:
(a) t is added: then you've found an augmenting path: go backwards from t along all
vertices based on "who brought them into LABEL". Increase the flow.
(b) You've scanned all vertices and no one else can be brought in. I claim you're at a max
flow.
Hopefully useful analogy:
Imagine s is
"patient 0"
(
a visual image
) for a disease. We're
curious if "t" (YOU) will be infected. A person connected to s becomes infected if there's
a contact with him that is unsaturated (perhaps b(x,y)f(x,y) is "contact time"). Looking at
people not yet infected, we ask if there's an unsaturated connection with those that are
infected. If yes, they become infected too etc.
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 Spring '08
 Staff
 Math, Bipartite graph, Ford Fulkerson, d. Scaling, FordFulkerson

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