25 Aug The Randomized max-cut problem is to find an efficient randomized algorithm which takes as input a graph G and which outputs a cut (S,T) of G
1. The Randomized max-cut problem is to find an efficient randomized algorithm which takes as input a graph G and which outputs a cut (S,T) of G. In this problem we also want to compare this algorithm to the 2-approximation for max cut given in class.
Here is the randomized algorithm for constructing a cut C = (S,T) in a graph G:
Go through the list of vertices v1, v2, v3, …, vn in G. For each vertex vi flip a fair coin and if it comes up head put the vertex in S, otherwise put it in T.
Answer questions i., ii. and iii. below about this randomized algorithm:
(i). What is the expected value of |C| = |(S, T)| constructed above?
Your answer should show the expected value of the cut |C|, in terms of the number m of edges
in G. You should briefly explain how you calculated this expected value and what properties of
expected value you used to obtain your answer.
(ii). Compare this algorithm to the 2-approximation for maxcut given in class.
Specifically, does it give stronger result or a weaker result (or an incomparable) result to the
deterministic algorithm given in class? Briefly explain why you think that.
2. This problem is the same as problem 1, only for a weighted graph. You still use the same algorithm as given before part (i) in problem 2 above. The weight of the
cut you get here is the sum of the weights of the edges across the cut.
What is the expected value of the weighted cut (S,T) constructed above ?
This question is pretty much the same as before as well, but now the size of the cut is the sum of
the weights of the edges crossing the cut.
Your answer should give the expected value in terms of the weights of the edges of G.
Again you should show how you calculated this expected value.
Do you think this gives us a 2-approximation for the weightd max cut problem ? Explain why
or why not ?
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