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Question: exercise 8 maxcut the probabilistic method is a very useful...

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Exercise 8 (MAX-CUT). The probabilistic method is a very useful way to prove the ex- istence of something satisfying some properties. This method is based upon the following elementary statement: If Q E R and if a random variable X : Ω → R satisfies EX-α, then there exists some w E Ω such that X(w)-a. We will demonstrate this principle in this exercise Let G- (V, E) be an undirected graph on the vertices V 1,... ,n] so that the edge set E is a subset of unordered pairs {i, j} such that i, j EV and i fj. Let SCV and denote Sc := V 〈 S. We refer to (S, Sc) as a cut of the graph G. The goal of the MAX-CUT problem is to maximize the number of edges going between S and SC over all cuts of the graph G. Prove that there exists a cut (S, Se) of the graph such that the number of edges going between S and S is at least E| /2. (Hint: define a random S C V such that, for every i E V. Pi E S) = 1/2, and the events 1 E S, 2 E S, , n E S are all independent. If {i.Λ e E. show that P(i E S. j ¢ S) = 1/4. So, what is the expected number of edges {i, j} E E such that i E S and J ¢ S?)

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