Duality

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Duality

• Lecture 10 Feb 9

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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. Weve 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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Example
Is optimal solution lt 30?
Yes, consider (2,1,3)
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Example
Upper bound is easy to prove, we just need to
give a solution.
This shows that the problem is in NP.
What about lower bounds?
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Example
Is optimal solution gt 5?
Yes, because x3 gt 1.
Is optimal solution gt 6?
Yes, because 5x1 x2 gt 6.
Is optimal solution gt 16?
Yes, because 6x1 x2 2x3 gt 16.
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Strategy
What is the strategy we used to prove lower
bounds?
Take a linear combination of constraints!
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Strategy
Dont reverse inequalities.
Whats the objective??
To maximize the lower bound.
Optimal solution 26
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Primal Dual Programs
Dual Program
Primal Program
Primal solutions
Dual solutions
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Weak Duality
Theorem
If x and y are feasible primal and dual
solutions, then
Proof
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Maximum bipartite matchings
To obtain best upper bound.
What does the dual program means?
Fractional vertex cover!
Maximum matching lt maximum fractional matching
lt minimum fractional vertex cover lt minimum
vertex cover
By Konig, equality throughout!
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Maximum Flow
d(i,j)1
t
s
What does the dual means?
pv 1
pv 0
Minimum cut is a feasible solution.
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Maximum Flow
Maximum flow lt maximum fractional flow lt
minimum fractional cut lt minimum cut
By max-flow-min-cut, equality throughout!
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Primal Dual Programs
Dual Program
Primal Program
Primal solutions
Dual solutions
In maximum bipartite matching and maximum
flow, The primal optimal solution the dual
optimal solution.
Example where there is a gap?
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Strong Duality
Example where there is a gap?
Never.
Von Neumann 1947
Primal optimal Dual optimal
Dual solutions
Primal solutions
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Strong Duality
PROVE
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Example
Objective max
2
1
-1
1
-2
2
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Example
Objective max
2
1
-1
1
-2
2
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Geometric Intuition
2
1
-1
1
-2
2
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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!
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Strong Duality
Intuition There exist Y1 y2 so that
Primal optimal value
Y (y1, y2) is the dual optimal solution!
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2 Player Game
Column player
-1
1
0
Strategy A probability distribution
Row player
0
-1
1
1
0
-1
Row player tries to maximize the payoff, column
player tries to minimize
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2 Player Game
Column player
Strategy A probability distribution
Row player
A(i,j)
okay!
You have to decide your strategy first.
Is it fair??
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Von Neumann Minimax Theorem
Strategy set
Which player decides first doesnt matter!
Think of paper, scissor, rock.
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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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Key Observation
similarly
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Primal Dual Programs
duality
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Chinese New Year
Homework discussion next Thursday.
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