Why it matters

Combines max flow + optimization. Understanding shapes assignment problems.

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The architecture

Successive shortest paths: augment via shortest-cost path each iteration.

Cycle cancellation: eliminate negative-cost cycles.

Min-cost flow approachesSuccessive shortest pathsBellman-Ford augmentCycle cancellationeliminate neg cyclesSPFA / Dijkstra with potentialsfaster in practiceComplexity: O(VE) per SPFA augmentation × O(V) iterations = O(V²E)
MCMF algorithms.
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How it works end to end

Bellman-Ford or SPFA finds shortest cost path.

Augment along it.

Repeat until no augmenting path.

Applications: assignment problem, min-cost transportation.