How to determine simplex time complexity (ie Max flow)

Question

Simplex algorithm is said to have exponential worst case time complexity. Yet it is still often used in practice. How can you determine the average time complexity for a certain

Answer 1

The average case complexity is rather difficult to analyze and it depends on the distribution of your linear program. I believe that it was worked out to be polynomial time under some common distributions. I currently cannot find the paper though.

EDIT: Yes, here are the sources:

Nocedal, J. and Wright, S. J. Numerical Optimization. New York: Springer-Verlag, 1999.

Forsgren, A.; Gill, P. E.; and Wright, M. H. "Interior Methods for Nonlinear Optimization." SIAM Rev. 44, 525-597, 2002.

I read it in the first book and apparently it was proven in a separate paper (Forsgren). You could find either in a university library.

Answer 2

If it is still interesting. Time complexity of simplex is O((n+m)*n).

n - number of variables.

m - inequality constraints.

Why? Because the number of iterations could be no more than n + m in case of n which is an upper bound on the numbers of vertices .

But this upper bound is exponential in n.