hw1_sol_vsr - Homework 1 Problems 4,5 and 6 Sudheer Vakati...

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Homework 1: Problems 4,5 and 6 Sudheer Vakati September 11, 2009 Problem 4 a) True All arc capacities are even. Divide all the arc capacities by 2. Find the maximum flow for the network. Mutiply the flow on each edge by 2. The resulting flow is a maximum flow for the original ( even capacities on each arc) network. Maximum flow is even. Reason: Ford-Fulkersons algorithm gives a maximum flow which only has in- teger values. Mutiplying integer values by 2 gives even values. So the maximum flow is even. b) False Counter Example: Consider the following network with the vertex set V = s, 1, 2, 3, t The arcs and the arc capacities are as follows: (s, 1) = 3 (s, 2) = 3 (1, 3) = 3 (2, 3) = 3 (3, t) = 7 The maximum flow for this network is as follows: (s, 1) = 3 (s, 2) = 3 (1, 3) = 3 (2, 3) = 3 (3, t) = 6 It can be seen that the maximum flow is not odd Problem 5 Arcs from (A-A’) to (A’-A) and (A’-A) to (A-A’) will influence the mentioned inequality. The vertex set V can be divided into following mutual exclusive sets A-A’, A’-A, A A’, V- (A A’).
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