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In a flow network, suppose we add constraints of the following type:

The flow entering a vertex $v$ must be at most the flow exiting a vertex $u$.

Is maximum-flow with such constraints still solvable in polynomial time?

Is it possible to reduce it to a standard maximum-flow problem?

NOTE: I found a recent paper that studies network flow with constraints, however, the constraints studied there are different:

"A negative disjunctive constraint states that a certain pair of arcs cannot be simultaneously used for sending flow"

"A positive disjunctive constraint forces that for certain pairs of arcs at least one arc has to carry flow".

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First a quick note, the flow entering a vertex is equal to the flow leaving. I'll just refer to it as the amount of flow through a vertex.

Second, note that we can say $a \leq b \wedge b \leq a$, thus we can add the constraint $a = b$ for two vertices.

Then we can re-use the NP-hard result from the paper that you quoted:

"A negative disjunctive constraint states that a certain pair of arcs cannot be simultaneously used for sending flow"

The reduction is done by putting two proxy vertices in the middle of the respective arcs, let's call them $a$ and $b$.

We then also add two vertices $a'$ and $b'$, directly connecting to the source, and add the constraints $a = a'$ and $b = b'$. Finally we add vertex $c$, and connect it to $a'$, $b'$ and the sink. Note that the flow through $c = a' + b' = a + b$.

We finish the reduction by adding constraints $c \leq a$ and $c \leq b$. Now $a+b \leq a$ and $a+b \leq b$ implies that either $a=0$ or $b=0$ (or both).

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