1. CS 312: Algorithm Design & Analysis Lecture #29: Network Flow and Cuts This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, with contributions from Mike Jones

2. Announcements  HW #20:  Due now.  HW #20.5  Due Monday  Project #6: Linear Programming  In preparation: HW #21  Formulate problem for Proj. #6  NO 2-hour time limit  Due Wednesday

3. Objectives  Better understand the idea of “changing the problem” in the max flow algorithm  Understand the relationship between flows and cuts  Solve Bipartite Matching  See example of Padded matrix-vector representation

4. Max Flow Example s b d t 42 24 4 e f 2 4 6 Summarize the max flow algorithm.

5. Max Flow Example 1.Pick a simple path s  t and increase the flow along that path as much as possible 2.Create the residual graph G R including reverse edges 3. Repeat until there is no path with remaining capacity. s b d t 42 24 4 e f 2 4 6

6. Max Flow Example 1.Pick a simple path s  t and increase the flow along that path as much as possible 2.Create the residual graph G R including reverse edges 3. Repeat until there is no path with remaining capacity. s b d t 42 24 4 e f 2 4 6 Turn to your partner and tell them about a question you have about how this works. Remember / write your partner’s question.

7. Solve s b d t 42 24 4 e f 2 4 6

8. Max Flow Example s b d t 42 24 4 e f 2 6 s b d t e f Residual Graph, incl. reverse edges One solution 4

9. Max Flow Example s b d t e f s b d t e f Residual Graph, incl. reverse edges No paths from s to t, so we are done.

10. Max Flow Example s b d t 42 24 4 e f 2 6 s b d t e f Residual Graph, incl. reverse edges One solution 4

11. Max Flow Example s b d t e f s b d t e f Residual Graph, incl. reverse edges No paths from s to t, so we are done.

12. Max Flow Example s b d t e f s b d t e f Residual Graph, incl. reverse edges

13. Max Flow Example s b d t 42 24 4 e f 2 4 6 s b d t e f Residual Graph, incl. reverse edges Another solution

14. Max Flow Example s b d t e f s b d t e f Residual Graph, incl. reverse edges

15. Max Flow Example s b d t e f s b d t e f Residual Graph, incl. reverse edges No paths from s to t, so we are done.

16. Key Points about our Max Flow Algorithm  When we transform the problem we create “imaginary edges”  However, in the end our selected paths always produce a flow schedule that involves only the original edges.

17. (s,t)-Cut An (s,t)-cut partitions the vertices into two disjoint sets: L contains s R contains t s b d t 42 24 4 e f 2 4 6 To compute the capacity of a cut, add up the capacities of edges that flow across the cut from L to R. L: Source Side R: Sink Side

18. (s,t)-Cut An (s,t)-cut partitions the vertices into two disjoint sets: L contains s R contains t s b d t 42 24 4 e f 2 4 6 To compute the capacity of a cut, add up the capacities of edges that flow across the cut from L to R. L: Source Side R: Sink Side

19. Cut Property s b d t 42 24 4 e f 2 4 6 Max flow is less than or equal to the flow across any cut. Max-Flow Min-Cut Theorem: The maximum flow through a network is equal to the minimum flow across all cuts. What does a min cut represent?

20. Cut Property s b d t 42 24 4 e f 2 4 6 Max flow is less than or equal to the flow across any cut. Max-Flow Min-Cut Theorem: The maximum flow through a network is equal to the minimum flow across all cuts. What does a min cut represent?

21. Cut Property s b d t 42 24 4 e f 2 4 6 We can call a minimum cut a “matching cut”, since its capacity matches the maximum flow. Max flow is less than or equal to the flow across any cut. Max-Flow Min-Cut Theorem: The maximum flow through a network is equal to the minimum flow across all cuts.

22. Algorithms

23. Match-making

24. Is there a perfect matching?

25. Bipartite Matching Is there a perfect matching? Great example of problem reduction.

26. LP: Padded Matrices & Vectors

27. End

28. Assignment  HW #20.5  Due Monday  HW #21:  Part 1 of Project #6  Just the problem formulation and representations  No time limit  Due Wednesday

29. Another Example