CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Discussion Notes #10

from Wednesday 19 Nov 2003

Finding Maximum Flows

Please answer the questions during the discussion period.

Our analysis of max flows and min cuts last week leads to an algorithm (called the Ford-Fulkerson method) for finding max flows in a flow diagram. The residual graph for a flow is the flow diagram that labels each edge with the capacity in each direction, if any, not used by the flow. For example, if an edge can take from 0 to 6 units in one direction and 4 are currently going across it, the residual graph shows 2 units in the forward direction and 4 in the backward direction.

An augmenting path is a path of non-zero edges in the residual graph. If there is an augmenting path, we can increase the flow along it by an amount equal to its lowest-labeled edge. (If there is no augmenting path, then there is a cut equal to the flow capacity and the flow is maximum.) The Ford-Fulkerson method is to keep finding an augmenting path and increasing the flow as long as you can. It always works on integer flows, but not always quickly:

           1000       1000
         ------>(2)------
        /        ^       \
       /         |        \  
      /          |         v
   >(1)         1|        ((4))
      \          |         ^
       \         |        / 
        \  1000  |  1000 /
         ------>(3)------

• Question 1: Consider the flow diagram above. Explain how starting from a zero flow, we can find 2000 augmenting paths in succession, of capacity 1 each, before finding the optimal flow of 2000 from vertex 1 to vertex 4.

• Question 2: The Edmonds-Karp algorithm is to employ Ford-Fulkerson but always use an augmenting path that contains as few edges as possible. How do you find such a path, and how long does it take?

• Question 3: An augmenting path fills an edge if after the flow is increased by the path, the edge is at its capacity. In [CLRS] they prove that using Edmonds-Karp, no edge can be filled more than n times before the flow is maximum (n is the number of vertices and e the number of edges). Using this fact and the answer to Question 2, find a big-O bound on the running time of Edmonds-Karp.

• Question 4: The maximum matching problem is to input a bipartite graph and find a subset of the edges such that (a) no two edges share a vertex, and (b) the subset is as large as possible. Explain how to reduce the maximum matching problem to the max-flow problem.

Last modified 1 December 2003