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1 LP Duality Lecture 13: Feb 28

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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.

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

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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.

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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.

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

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

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

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

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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!

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

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

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13 Primal Program Dual Program Dual solutions Primal solutions Primal Dual Programs Dual solutionsPrimal solutions Von Neumann [1947] Primal optimal = Dual optimal

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14 Strong Duality PROVE:

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15 Fundamental Theorem on Linear Inequalities

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16 Proof of Fundamental Theorem

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17 Farkas Lemma

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18 Strong Duality PROVE:

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19 Example 2 1 1-22 Objective: max

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20 Example 2 1 1-22 Objective: max

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21 Geometric Intuition 2 1 1-22

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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!

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

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24 2 Player Game 0 1 1 0 10 Row player Column player Row player tries to maximize the payoff, column player tries to minimize Strategy: A probability distribution

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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??

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

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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

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28 Key Observation similarly

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29 Primal Dual Programs duality

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