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Toggle## Cycle-canceling algorithm

The cycle-canceling algorithm begins with a resolution of maximum flow. Then, iteratively, the algorithm searches for a cycle with a negative cost (negative cost if the edge is taken in the opposite direction to the flow) and increases the flow on this cycle. When there are no more negative cycles, the algorithm ends.

Example :

let's build it graph gap and look for a negative cycle:

The 4-2-3-4 cycle has a cost of -3 + 1 + 1 = -1, we can saturate it with a flow of 2.

Similarly with the 4-2-1-3-4 cycle with a cost of -3 -2 +2 +1 = -2, we can saturate it with a flow of 1.

There is no longer a negative cycle, the algorithm ends.