Unformatted text preview: Which of the following statements are true and which are false? Justify your answer. 1. If all directed edges in a network have distinct capacities, then there is a unique max ﬂow. 2. Consider a graph G = ( V,E ). Now, for each edge e = ( u,v ) with capacity c ( e ), we will add an edge e = ( v,u ) in the opposite direction with the same capacity c ( e ). This alteration to G will not change the value of the max ﬂow. Problem 4 . [Randomized Max Cut] Consider the Max Cut problem: given an undirected graph G = ( V,E ) and weight function w : E → Z + , ﬁnd a cut ( A,B ) such that the value of the weights across the cut is maximized . We will now analyze a simple randomized algorithm for this problem: For each v , independently put it in A with probability 1/2. Output the cut ( A,V \ A ). 1. What is the probability of edge ( u,v ) being in the cut? 2. What is the expected weight of the edges in the cut? 3. What is the maximum weight of any cut? 1...
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 Fall '08
 Chekuri,C
 Algorithms, Dynamic Programming, Graph Theory, Greedy algorithm, optimal solution

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