Fractional (LP)
Solvable by LP in polynomial time. Each commodity's flow variables + capacity constraints.
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Integral
NP-hard. Even 2 commodities. Approximation via LP rounding.
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Max concurrent
Maximize λ such that (1+λ)·demand routable for all commodities. Different objective.
Duality
Max multi-commodity flow ≠ min multi-cut in general. Gap of O(log k). Sparse cuts.