Edmonds-Karp

General Information

Algorithmic problem: Max-Flow Problems

Algorithm : This is a minor variation of Ford-Fulkerson: Among all flow-augmenting [math]\displaystyle{ (s,t) }[/math]-paths, always choose one with smallest number of arcs.

Abstract View

Invariant: After [math]\displaystyle{ i \ge 0 }[/math] iterations:

1. The flow [math]\displaystyle{ f }[/math] is a fleasible flow.
2. If all upper bounds are integral, [math]\displaystyle{ f }[/math] is integral as well.

Notation: For an [math]\displaystyle{ (s,t) }[/math]-flow, let [math]\displaystyle{ A_f }[/math] denote the set of all arcs that belong to at least one flow-augmenting [math]\displaystyle{ (s,t) }[/math]-path with smallest number of arcs.

Variant:

1. The smallest number of arcs on a flow-aumenting [math]\displaystyle{ (s,t) }[/math]-path increases (non-strictly) monotonously.
2. Whenever that number does not decrease in an iteration, the size of [math]\displaystyle{ A_f }[/math] decreases.

Break condition: There is no flow-augumenting path.

Complexity

Statement: Even if the upper bounds are not integral, the asymptotic complexity is in [math]\displaystyle{ \mathcal{O}(nm^2) }[/math], where [math]\displaystyle{ n=|V| }[/math] and [math]\displaystyle{ m=|A| }[/math].

Proof: If the variant is fulfilled, the smallest number of arcs on a flow-augmenting path strictly increases after at most [math]\displaystyle{ m }[/math] iterations. This number is positive, but cannot be larger than [math]\displaystyle{ n-1 }[/math]. Hence, the total number of iterations is in [math]\displaystyle{ \mathcal{O}(nm) }[/math].

Therefore, it suffices to show that the variant is fulfilled.