Dinic

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General Information

Algorithmic problem: Max-flow problem (standard version)

Type of algorithm : loop.

Abstract View

Invariant: After iterations:

  1. The flow is feasible.
  2. 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.

Induction basis

Abstract view: Initialize as an arbitrary feasible flow, for example, the zero flow.

Proof: Nothing to show.

Induction step

Abstract view:

  1. 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).
  2. Use one of the algorithms for the blocking flow problem to construct a blocking flow in with respect to the residual capacities for .
  3. Add to .

Proof: Obvious.

Correctness

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.

Complexity

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.