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Assume that, for a given network, its multi-sink multi-source max-flow satisfies a lexicographic order. Is this enough to state that I can solve the max-flow problem, by using a lex-max solver (i.e. a solver for the lexicographic maximal flow problem)?

If yes, I just need to prove my assumption. Namely, proving that the optimal solution for my max-flow problem has lexicographic order.

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