1. The Method of Conditional Probabilities
presented by Kwak, Nam-ju
Applied Algorithm Laboratory
24 JAN 2010

2. Table of Contents A Starting Example Generalization
Pessimistic Estimator
Example of Pessimistic Estimator

3. A Starting Example
Proposition: For every integer n there exists a coloring of the edges of the complete graph Kn by two colors so that the total number of monochromatic copies of K4 is at most .

4. A Starting Example Kn Is it monochromatic? K4 There are K4’s in a Kn.
A K4 is monochromatic with a probability of 2-5.

5. A Starting Example
Proposition: For every integer n there exists a coloring of the edges of the complete graph Kn by two colors so that the total number of monochromatic copies of K4 is at most .
the expected number of monochromatic copies of K4 in a random 2-edge-coloring of Kn.
This proposition says that a coloring for a Kn exists, such that it has at most monochromatic copies of K4.

6. A Starting Example
Let us color a Kn so that it may have at most monochromatic K4’s.
Such a coloring can hopefully be found in polynomial time in terms of n, deterministically.
RED and BLUE are used for coloring.

7. A Starting Example K: each copy K4 of Kn w(K): given a K4, namely K…
at least 1 edge is colored red and at least 1 edge is colored blue : w(K)=0
0 edge is colored: w(K)=2-5
r edges are colored, all with the same color, where r≥1: w(K)=2r-6 W=

8. A Starting Example Coloring strategy
Color each edge of Kn in turn. It will be finished in n(n-1)/2 stages.
Assume that, at a given stage i, a list of edges e1, …, ei-1 have already been colored.
Then, we should color ei, right now.

9. A Starting Example Coloring strategy
Wred, Wblue: the value of W after coloring ei red and blue, respectively.
W=(Wred+Wblue)/2
Color ei red if Wred≤Wblue, blue otherwise.
Then, W never increase for all the stages.

10. A Starting Example Coloring strategy
Since W is non-increasing and the initial value is , the final value of W is less than equal to it.
The final value of W (after coloring all the edges) is the actual # of monochromatic K4’s of Kn.

11. Generalization
A1, …, As: events
ϵ1, …, ϵq: binary, q stages

12. Pessimistic Estimator
There are cases for which the previous approach does not work well.
Under the following 2 conditions,
We can say

13. Example of Pessimistic Estimator
Theorem: Let be an n by n matrix of reals, where -1≤aij≤1 for all i, j. Then one can find, in polynomial time, ϵ1, …, ϵn∈{-1, 1} such that for every i, 1≤i≤n,

14. Example of Pessimistic Estimator
Ai: the event
α=β/n
Since ,
We define pessimistic estimators

15. Example of Pessimistic Estimator
We should show
In addition, .
Of course, those claims can be proved, however, here the proofs are skipped.

16. Conclusion
Here, we learnt a way to extract deterministic information from randomized approaches.

17. Q&A Ask questions, if any, please.
The contents are based on chapter 16.1 of The Probabilistic Method, 3rd ed., written by Noga Alon and Joel H. Spencer.