Algorithmic problem: Max-flow problem (standard version)
Type of algorithm : loop.
Invariant: After iterations:
- The flow is feasible.
- If all upper bounds are integral, the -values are integral as well.
Variant: The smallest number of arcs on a flow-augmenting -path strictly increases.
Break condition: There is no flow-augmenting -path anymore.
Abstract view: Initialize as an arbitrary feasible flow, for example, the zero flow.
Proof: Nothing to show.
- Construct the acyclic subgraph of the residual network that contains an arc if, and only if, the arc is on at least one -path with smallest number of arcs (and contains all nodes incident to these arcs).
- Use one of the algorithms for the blocking flow problem to construct a blocking flow in with respect to the residual capacities for .
- Add to .
Feasibility of follows immediately from the invariant. If the algorithm terminates, the break condition immediately proves maximality along with the max-flow min-cut theorem. Termination follows immediately from the following complexity considerations.
Statement: The asymptotic complexity is in , where , , and is the asymptotic complexity of the blocking-flow algorithm.
Proof: Evidenty, the smallest number of arcs on a flow-augmenting -path cannot exceed . Therefore, the variant implies that the algorithm terminates after iterations. The complexity of a single iteration is dominated by the computation of a blocking flow.