Download presentation

Presentation is loading. Please wait.

Published byDana Grimmett Modified over 3 years ago

1
1 LP Duality Lecture 13: Feb 28

2
2 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. 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.

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

4
4 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.

5
5 Example 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.

6
6 Strategy What is the strategy we used to prove lower bounds? Take a linear combination of constraints!

7
7 Strategy Don’t reverse inequalities. What’s the objective?? To maximize the lower bound. Optimal solution = 26

8
8 Primal Dual Programs Primal Program Dual Program Dual solutions Primal solutions

9
9 Weak Duality If x and y are feasible primal and dual solutions, then Theorem Proof

10
10 Maximum 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!

11
11 Maximum Flow s t What does the dual means? pv = 1 pv = 0 d(i,j)=1 Minimum cut is a feasible solution.

12
12 Maximum Flow Maximum flow <= maximum fractional flow <= minimum fractional cut <= minimum cut By max-flow-min-cut, equality throughout!

13
13 Primal Program Dual Program Dual solutions Primal solutions Primal Dual Programs Dual solutionsPrimal solutions Von Neumann [1947] Primal optimal = Dual optimal

14
14 Strong Duality PROVE:

15
15 Fundamental Theorem on Linear Inequalities

16
16 Proof of Fundamental Theorem

17
17 Farkas Lemma

18
18 Strong Duality PROVE:

19
19 Example 2 1 1-22 Objective: max

20
20 Example 2 1 1-22 Objective: max

21
21 Geometric Intuition 2 1 1-22

22
22 Geometric Intuition Intuition: There exist nonnegative Y1 y2 so that The vector c can be generated by a1, a2. Y = (y1, y2) is the dual optimal solution!

23
23 Strong Duality Intuition: There exist Y1 y2 so that Y = (y1, y2) is the dual optimal solution! Primal optimal value

24
24 2 Player Game 0 1 1 0 10 Row player Column player Row player tries to maximize the payoff, column player tries to minimize Strategy: A probability distribution

25
25 2 Player Game A(i,j) Row player Column player Strategy: A probability distribution You have to decide your strategy first. Is it fair??

26
26 Von Neumann Minimax Theorem Strategy set Which player decides first doesn’t matter! e.g. paper, scissor, rock.

27
27 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

28
28 Key Observation similarly

29
29 Primal Dual Programs duality

Similar presentations

Presentation is loading. Please wait....

OK

Linear Programming – Max Flow – Min Cut Orgad Keller.

Linear Programming – Max Flow – Min Cut Orgad Keller.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on track feed battery charger Ppt on computer ethics crime and security Ppt on diodes and transistors Ppt on circuit switching network Ppt on health care system Ppt on multiplication for class 2 Ppt on grease lubrication for o-rings Ppt on mars one astronauts Ppt on email etiquettes presentation rubric Ppt on 7 ages by william shakespeare