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Anyone recognize this as a special type of multi-commodity flow problem?
I think you can solve this as a single shortest path problem with respect to arc weights
$$\overline{w}_{ij}=\min\{-w_{ijp}\ :\ p\in\{1,\ldots,m\}\}.$$
For every arc $(i,j)$ fix some $p(i,j)\in\{1,\...
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