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
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
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
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