LP Duality
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Transcript of LP Duality
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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
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-1
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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
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-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