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If you want to maximize a function f(x), you can do this by minimizing -f(x). Naively it seems like an analogous trick could convert a max cut problem into a min cut problem, however this is apparently not possible because min cut is easy while max cut is hard. What is the clearest explanation of how this naive intuition goes wrong?

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The ease of the minimum cut selection depends on the edge capacities being non-negative. If you negate all the capacities, that assumption goes out the window. And there is no positive edge function you can add to reinstate it.

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If you try to take the complement of the edge set to translate a max cut problem, you end up not minimizing cuts, but cuts minus a correction factor that depends on the size of the subsets in the partition (namely, it is the product of the sizes). This changes the nature of the problem considerably - if you have a dense graph, min cut will tend to separate a small chunk, while the corrected version will not.

Edit: A previous version complained about a difference in input type, hence David's comment.

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It's easy to find a global min cut (without specified source and sink) in polynomial time, for instance by testing all possible source-sink pairs. – David Eppstein Dec 9 '09 at 19:19

see also: shortest path (Dijkstra/bfs) and longest path (NP-hard) in a graph

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This analogy can be made rigorous: min cut and shortest path are dual problems in planar graphs, for which max cut and longest path are still hard. – David Eppstein Dec 10 '09 at 17:59

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