Network Flow Summary

First question:
In the cost flow, on the premise that the source image does not have a negative ring, why does the shortest path of augmented reality not fall into a negative ring?

In fact, this is a very simple problem, but I have struggled for a long time, 233.

First, assume that the residual graph will have a negative ring, and the cause of the negative ring is that some reverse arcs are added to the residual graph after the augmented reality.

However, the augmented path must have no loops, so these reverse arcs may only be part of the negative ring.

Set the paths composed of the reverse arc to P, the paths composed of the edges corresponding to the reverse arc on P to P, and the paths composed of the other parts of the negative ring to Q.

The premise that p becomes a part of the negative ring is that the weight and absolute value of P are greater than the weight and of Q.

The premise is that the weight of P and the sum of the weight greater than Q.

Obviously, the above premise is in conflict with our search for the shortest increase path of the weight value.

If the above premise is true, we will get a shorter augmented path about the weight value by replacing Q with P.

Therefore, on the premise that the source image does not have a negative ring, the augmented residual graph will not have a negative ring.