The minimum cost flow problem (MCFP) is important because a huge number of problems are special cases of it. We saw in 8.5 that the longest-path problem can be viewed as a MCFP and leads to the critical path method, one of the oldest of all operational research techniques.

We have studied the solution of the MCFP by the network simplex algorithm. As with a general linear programming problem, the possibility of making a bad choice of pivots can give the algorithm a worse that polynomial running time. However, there do exist simplex algorithms which do run in polynomial time. See

James B. Orlin (1997). "A polynomial time primal network simplex algorithm for minimum cost flows". Mathematical Programming 78: 109–129.

Suppose $n$ and $m$ are the numbers of vertices and edges, respectively, and each cost is either $\infty$ or an integer bounded by $C$. Orlin describes an simplex-like pivoting algorithm with running $O\bigl(\min(n^2m \log nC, n^2m^2 \log n)\bigr)$.

We have studied the solution of the MCFP by the network simplex algorithm. As with a general linear programming problem, the possibility of making a bad choice of pivots can give the algorithm a worse that polynomial running time. However, there do exist simplex algorithms which do run in polynomial time. See

James B. Orlin (1997). "A polynomial time primal network simplex algorithm for minimum cost flows". Mathematical Programming 78: 109–129.

Suppose $n$ and $m$ are the numbers of vertices and edges, respectively, and each cost is either $\infty$ or an integer bounded by $C$. Orlin describes an simplex-like pivoting algorithm with running $O\bigl(\min(n^2m \log nC, n^2m^2 \log n)\bigr)$.