1 LP Duality Lecture 13: Feb 28

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

5 Example Is optimal solution >= 5? Yes, because x3 >= 1. 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 Primal Program Dual Program Dual solutions Primal solutions

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

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

11 Maximum Flow s t What does the dual means? pv = 1 pv = 0 d(i,j)=1 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 Program Dual Program Dual solutions Primal solutions Primal Dual Programs Dual solutionsPrimal solutions Von Neumann [1947] Primal optimal = Dual optimal

14 Strong Duality PROVE:

15 Fundamental Theorem on Linear Inequalities

16 Proof of Fundamental Theorem

17 Farkas Lemma

18 Strong Duality PROVE:

19 Example Objective: max

20 Example Objective: max

21 Geometric Intuition

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

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

24 2 Player Game Row player Column player Row player tries to maximize the payoff, column player tries to minimize Strategy: A probability distribution

25 2 Player Game A(i,j) Row player Column player Strategy: A probability distribution You have to decide your strategy first. Is it fair??

26 Von Neumann Minimax Theorem Strategy set Which player decides first doesn’t matter! e.g. paper, scissor, rock.

27 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

28 Key Observation similarly

29 Primal Dual Programs duality