Duality

1. Duality Lecture 10: Feb 9

2. Min-Max theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum Cut Both these relations can be derived from the combinatorial algorithms. We’ve also seen how to solve these problems by linear programming. Can we also obtain these min-max theorems from linear programming? Yes, LP-duality theorem.

3. Example Is optimal solution <= 30? Yes, consider (2,1,3)

4. Example Upper bound is easy to “prove”, we just need to give a solution. This shows that the problem is in NP. What about lower bounds?

5. Example Yes, because x3 >= 1. Is optimal solution >= 5? Is optimal solution >= 6? Yes, because 5x1 + x2 >= 6. Is optimal solution >= 16? Yes, because 6x1 + x2 +2x3 >= 16.

6. Strategy What is the strategy we used to prove lower bounds? Take a linear combination of constraints!

7. Strategy Don’t reverse inequalities. What’s the objective?? To maximize the lower bound. Optimal solution = 26

8. Primal Dual Programs Dual Program Primal Program Primal solutions Dual solutions

9. Weak Duality Theorem If x and y are feasible primal and dual solutions, then Proof

10. Maximum bipartite matchings To obtain best upper bound. Fractional vertex cover! What does the dual program means? Maximum matching <= maximum fractional matching <= minimum fractional vertex cover <= minimum vertex cover By Konig, equality throughout!

11. Maximum Flow d(i,j)=1 t s What does the dual means? pv = 1 pv = 0 Minimum cut is a feasible solution.

12. Maximum Flow Maximum flow <= maximum fractional flow <= minimum fractional cut <= minimum cut By max-flow-min-cut, equality throughout!

13. Primal Dual Programs Dual Program Primal Program Primal solutions Dual solutions In maximum bipartite matching and maximum flow, The primal optimal solution = the dual optimal solution. Example where there is a gap?

14. Strong Duality Example where there is a gap? Never. Von Neumann [1947] Primal optimal = Dual optimal Dual solutions Primal solutions

15. Strong Duality PROVE:

16. Example Objective: max 2 1 -2 -1 1 2

17. Example Objective: max 2 1 -2 -1 1 2

18. Geometric Intuition 2 1 -2 -1 1 2

19. Geometric Intuition Intuition: There exist Y1 y2 so that The vector c can be generated by a1, a2. Y = (y1, y2) is the dual optimal solution!

20. Strong Duality Intuition: There exist Y1 y2 so that Primal optimal value Y = (y1, y2) is the dual optimal solution!

21. 2 Player Game Column player -1 1 0 Strategy: A probability distribution Row player 0 -1 1 1 0 -1 Row player tries to maximize the payoff, column player tries to minimize

22. 2 Player Game Column player Strategy: A probability distribution Row player A(i,j) okay! You have to decide your strategy first. Is it fair??

23. Von Neumann Minimax Theorem Strategy set Which player decides first doesn’t matter! Think of paper, scissor, rock.

24. Key Observation If the row player fixes his strategy, then we can assume that y chooses a pure strategy Vertex solution is of the form (0,0,…,1,…0), i.e. a pure strategy

25. Key Observation similarly

26. Primal Dual Programs duality

27. Chinese New Year Homework discussion next Thursday. Please sign up project meeting.