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== Induction step ==
== Induction step ==
# Construct the [[Basic graph definitions|subgraph]] <math>G'=(V,A')</math> of the residual network that contains an arc if, and only if, the arc is on at least one <math>(s,t)</math>-path with smallest number of arcs.
# Construct a blocking flow <math>f'</math> in <math>G'</math> wth respect to the residual capacities for <math>f</math>.
For all arcs <math>a\in A</math>, add <math>f'</math> to <math>f</math>.

Revision as of 13:42, 13 October 2014

General Information

Algorithmic problem: Max-Flow Problems

Type of algorithm : loop.

Abstract View

Invariant: After [math]\displaystyle{ i \ge 0 }[/math] iterations: The flow [math]\displaystyle{ f }[/math] is feasible. If all upper bounds are integral, the [math]\displaystyle{ f }[/math]-values are integral as well.

Variant: The smallest number of arcs on a flow-augmenting [math]\displaystyle{ (s,t) }[/math]-path strictly increases.

Induction basis

Abstract view: Initialize [math]\displaystyle{ f }[/math] as an arbitrary feasible flow, for example, the zero flow.

Proof: Nothing to show.

Induction step

  1. Construct the subgraph [math]\displaystyle{ G'=(V,A') }[/math] of the residual network that contains an arc if, and only if, the arc is on at least one [math]\displaystyle{ (s,t) }[/math]-path with smallest number of arcs.
  2. Construct a blocking flow [math]\displaystyle{ f' }[/math] in [math]\displaystyle{ G' }[/math] wth respect to the residual capacities for [math]\displaystyle{ f }[/math].
For all arcs [math]\displaystyle{ a\in A }[/math], add [math]\displaystyle{ f' }[/math] to [math]\displaystyle{ f }[/math].