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

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

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 .

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.

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.

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.

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=

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.

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.

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.

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

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

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,

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

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

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

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.