The Method of Conditional Probabilities

1. The Method ofConditional 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 eired and blue, respectively. • W=(Wred+Wblue)/2 • Color eired 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 NogaAlon and Joel H. Spencer.