Out-of-Kilter algorithm

Out-of-kilter algorithm

Also called the Fulkerson-Ford algorithm (not to be confused with the maximum flow algorithm), the out-of-kilter algorithm allows to calculate a maximum flow at minimum cost with min and max bounds on the edges.

We will say that an edge is In-Kilter if it satisfies the complementary deviations, otherwise we will say that it is Out-of-Kilter (kilter means "in good condition").

The following explanations will not be translated for the sake of understanding the underlying logic in the sense of "kilter".

Theorem: A feasible solution x * is an optimal solution of the MCF problem if and only if for some set of node potentials π, the reduced costs and flow values satisfy the following complementary slackness conditions for every edge (u, v) in the network:

out-of-kilter algorithm

Which gives the following efficiency diagram:

out-of-kilter algorithm
out-of-kilter algorithm
out-of-kilter algorithm
out-of-kilter algorithm

Correctness:

  • Kilter numbers of the edges are non-increasing
    • Two operations in the algorithm affect the kilter numbers of arcs:
      • updating node potentials
      • augmenting flow along the cycle W
    • At each iteration the algorithm selects and edge (p, q)
      • Makes it in-kilter during potential update
      • decrease it by at least 1 by the flow increase

The algorithm terminate within O (mU) iterations, total runtime is: O (m ^ 2 U + mUn log n)

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