Suppose that you wish to find, among all minimum cuts in a flow network G withintegral capacities, one that contains the smallest number of edges. Show how tomodify the capacities of G to create a new flow network G0 in which any minimumcut in G0 is a minimum cut with the smallest number of edges in G.
Question
Suppose that you wish to find, among all minimum cuts in a flow network G withintegral capacities, one that contains the smallest number of edges. Show how tomodify the capacities of G to create a new flow network G0 in which any minimumcut in G0 is a minimum cut with the smallest number of edges in G.
Solution
To find a minimum cut with the smallest number of edges in a flow network G with integral capacities, you can modify the capacities of G to create a new flow network G0 as follows:
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First, assign a small fractional capacity ε (where 0 < ε < 1) to each edge in the network G. This will not change the integral capacities of the network but will add a small fractional capacity to each edge.
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Now, find a minimum cut in the new network G0. Since the capacities are now fractional, the minimum cut will prefer cuts with fewer edges because each edge contributes an additional ε to the total capacity.
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The minimum cut found in G0 is also a minimum cut in the original network G. This is because the total capacity of any cut in G0 is the same as the total capacity of the corresponding cut in G, except for the additional ε contributed by each edge. Therefore, if a cut is minimum in G0, it must also be minimum in G.
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Finally, note that the minimum cut found in G0 has the smallest number of edges among all minimum cuts in G. This is because any other minimum cut in G would have a larger total capacity in G0 due to the additional ε contributed by each edge.
Therefore, by modifying the capacities of G to create a new flow network G0, you can find a minimum cut in G that contains the smallest number of edges.
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