Presentation on theme: "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."— Presentation transcript:
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  Primal optimal = Dual optimal