(Latest Revision: Nov 23, 2021)

Solution to the 2022 Flow Network Problem

flow problem solution, page 01
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flow problem solution, page 03
flow problem solution, page 03
flow problem solution, page 03

For this problem, the A side of the minimum cut is {s, a}. No other nodes are reachable from s in the final version of the residual graph, Gf. The capacity of the cut is the sum of the capacities of the edges (in G) leaving the A side of the cut, 3 + 1 + 2 + 1 = 7. The value of the max flow is the sum of the flows on the edges leaving the source node s in G. That sum is 4 + 2 + 1, which also equals 7. These calculations verify that the max value of an s-t flow is equal to the min capacity of an s-t cut, which, according to the "Max-Flow, Min-Cut Theorem," is true for any flow network.