Duality Lecture 10: Feb 9

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.

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

Example 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.

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.

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

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

Primal Dual Programs Primal Program Dual Program Dual solutions Primal solutions

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

Maximum bipartite matchings 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!

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

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

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

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

Strong Duality PROVE:

Example Objective: max

Example Objective: max

Geometric Intuition

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!

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

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

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

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

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

Key Observation similarly

Primal Dual Programs duality

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