Duality

Lecture 10: Feb 9

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.

ExampleExample

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

ExampleExample

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.

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.

StrategyStrategy

What is the strategy we used to prove lower bounds?

Take a linear combination of constraints!

StrategyStrategy

Don’t reverse inequalities.

What’s the objective??

To maximize the lower bound.Optimal solution = 26

Primal Dual ProgramsPrimal Dual Programs

Primal Program Dual Program

Dual solutions Primal solutions

Weak DualityWeak Duality

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

Theorem

Proof

Maximum bipartite matchingsMaximum 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 FlowMaximum Flow

s tWhat does the dual means?

pv = 1 pv = 0

d(i,j)=1

Minimum cut is a feasible solution.

Maximum FlowMaximum 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 ProgramsPrimal Dual Programs

In maximum bipartite matching and maximum flow,

The primal optimal solution = the dual optimal solution.

Example where there is a gap?

Strong DualityStrong Duality

Never.

Example where there is a gap?

Von Neumann [1947]

Primal optimal = Dual optimal

Dual solutions Primal solutions

Strong DualityStrong Duality

PROVE:

ExampleExample

2

-1

1

1-2 2

Objective: max

ExampleExample

2

-1

1

1-2 2

Objective: max

Geometric IntuitionGeometric Intuition

2

-1

1

1-2 2

Geometric IntuitionGeometric Intuition

Intuition:There existY1 y2 so that

The vector c can be generated by a1, a2.

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

Strong DualityStrong Duality

Intuition:There existY1 y2 so that

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

Primal optimal value

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

2 Player Game2 Player Game

A(i,j)Row player

Column playerStrategy:A probabilitydistribution

You have to decide your strategy first.

Is it fair??

okay!

Von Neumann Minimax TheoremVon Neumann Minimax Theorem

Strategy set

Which player decides first doesn’t matter!

Think of paper, scissor, rock.

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

Key ObservationKey Observation

similarly

Primal Dual ProgramsPrimal Dual Programs

duality

Chinese New YearChinese New Year

Homework discussion next Thursday.

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