CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Discussion Notes #9

from Wednesday 12 Nov 2003

Min Cuts and Max Flows

Please answer the questions during the discussion period.

A flow diagram is a labeled directed graph like this example which I have gratefully lifted from www.ms.ic.ac.uk/jeb/or/netflow.html:


           3          5            3
         ------>(2)-------->(5)------
        /        ^ <------   |       \
       /        1|       4\  |2       \
      /   2      |    1    \ v    1    >
   >(1)-------->(3)-------->(6)-------->((8))
      \          ^           |         >
       \         |2          |2       /
        \   3    |     2     v      5/
         ------>(4)-------->(7)------

The edge labels represent the capacity of the edge. A flow is an assignment of an amount, from zero to the capacity, to each edge that has zero net flow in or out of each intermediate vertex. The max-flow problem is to input the diagram and output a flow with the maximum amount leaving the source node and entering the sink node.

A cut is a set of vertices including the source but not the sink. The capacity of a cut is the total capacity of all edges from vertices in the cut to those not in it. The min-cut is the cut of smallest size. The min-cut max-flow theorem says that the min-cut and max-flow always have the same size.

Question 1: Find the min-cut and the max-flow in the example above.
Question 2: How many cuts are there in a flow diagram with $n$ vertices? Is brute-force search for the min-cut a good general way to find the max-flow?
Question 3: Argue that the max-flow cannot have greater size than the min-cut.
Question 4: Argue that the min-cut cannot have greater size than the max-flow. (Hint: Assume we have a flow that is smaller than every cut. Prove that there is an augmenting path -- a path from the source to the sink that uses some positive amount of excess capacity on each edge. Then argue that this shows the given flow is not maximum.)

Last modified 12 November 2003