Why it matters

Real problems need flow extensions. Understanding shapes advanced modeling.

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The architecture

Vertex capacity: split v into v_in + v_out with edge of that capacity.

Lower bounds: transform to standard.

Flow extensionsVertex capacitysplit vLower boundsreduce to standardGomory-Hu treeall-pairs min cutsMulti-commodity flow: NP-hard in general
Advanced flow problems.
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How it works end to end

Gomory-Hu tree: n-1 min-cut computations to know all pairs. Compact representation.

Multi-commodity flow: multiple commodities compete for capacity. LP formulation.

Applications: transportation, network design.