The minimum cost flow problem

Simplifying assumptions Network is connected (as an undirected graph). – We can consider each connected component separately. Network is uncapacitated (no arc constraints). – We can transform any network to an equivalent uncapacitated network. – Alternatively: The network simplex algorithm can be extended to handle upper and lower bounds. The network has no 2-cycles – Only simplifies the exposition. The algorithm works also with 2-cycles.

Klein’s cycle cancelling algorithm

Cycle flows

Observations

Augmenting cycles

Klein’s algorithm.

The residual network Like the residual network used in the Ford- Fulkerson max-flow algorithm – represents possible changes to flow on arcs maintaining feasibility. Here the residual network will: – Have the same set of nodes. – Specify that feasible flows must be circulations. – Have upper bounds (a.k.a. residual capacities) and costs, but no lower bounds.

Definition of the residual network

Observations

Difference between feasible flows.

Adding feasible residual flow.

Finding a negative cycle

Partial correctness

Proof of partial correctness

Termination Like the Ford-Fulkerson max-flow algorithm, Klein’s cycle cancelling algorithm is not guaranteed to terminate! However if all balances, lower bounds, and upper bounds are integers and we start with an integer feasible flow we must terminate. Proof: Only finitely many feasible integer flows and we improve in each iteration.

Complexity Each iteration may be done efficiently. Even when guaranteed termination by integer input the algorithm may take exponential time. There exists a choice for selecting an augmenting cycle leading to a polynomial worst case running time! In practice: Hard to compete with optimized network simplex algorithm.

Finding the first feasible flow