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LP Duality

Lecture 13: Feb 28

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Min-Max TheoremsMin-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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ExampleExample

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

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NP and co-NP?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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ExampleExample

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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StrategyStrategy

What is the strategy we used to prove lower bounds?

Take a linear combination of constraints!

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StrategyStrategy

Don’t reverse inequalities.

What’s the objective??

To maximize the lower bound.Optimal solution = 26

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Primal Dual ProgramsPrimal Dual Programs

Primal Program Dual Program

Dual solutions Primal solutions

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Weak DualityWeak Duality

If x and y are feasible primal and dual solutions, then

Theorem

Proof

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Maximum bipartite matchingMaximum 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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Maximum FlowMaximum Flow

s tWhat does the dual means?

pv = 1 pv = 0

d(i,j)=1

Minimum cut is a feasible solution.

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Maximum FlowMaximum Flow

Maximum flow <= maximum fractional flow <=

minimum fractional cut <= minimum cut

By max-flow-min-cut, equality throughout!

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Primal Program Dual Program

Dual solutions Primal solutions

Primal Dual ProgramsPrimal Dual Programs

Dual solutions Primal solutions

Von Neumann [1947] Primal optimal = Dual optimal

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Strong DualityStrong Duality

PROVE:

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Fundamental Theorem on Linear InequalitiesFundamental Theorem on Linear Inequalities

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Proof of Fundamental TheoremProof of Fundamental Theorem

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Farkas LemmaFarkas Lemma

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Strong DualityStrong Duality

PROVE:

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ExampleExample

2

-1

1

1-2 2

Objective: max

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ExampleExample

2

-1

1

1-2 2

Objective: max

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Geometric IntuitionGeometric Intuition

2

-1

1

1-2 2

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Geometric IntuitionGeometric Intuition

Intuition:There exist nonnegativeY1 y2 so that

The vector c can be generated by a1, a2.

Y = (y1, y2) is the dual optimal solution!

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Strong DualityStrong Duality

Intuition:There existY1 y2 so that

Y = (y1, y2) is the dual optimal solution!

Primal optimal value

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2 Player Game2 Player Game

0 -1 1

1 0 -1

-1 1 0

Row player

Column player

Row player tries to maximize the payoff, column player tries to minimize

Strategy:A probabilitydistribution

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2 Player Game2 Player Game

A(i,j)Row player

Column playerStrategy:A probabilitydistribution

You have to decide your strategy first.

Is it fair??

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Von Neumann Minimax TheoremVon Neumann Minimax Theorem

Strategy set

Which player decides first doesn’t matter!

e.g. paper, scissor, rock.

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Key ObservationKey Observation

If the row player fixes his strategy,

then we can assume that y chooses a pure strategy

Vertex solutionis of the form(0,0,…,1,…0),i.e. a pure strategy

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Key ObservationKey Observation

similarly

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Primal Dual ProgramsPrimal Dual Programs

duality