How to Find the time complexity of an algorithm with mixed linear and polynomial times?
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am trying to work out empirically the complexity of an algorithm I wrote. I have two variables in the input: dataset size (dimensions of a matrix) and number of values computed.
When I look at the complexity varying the number of values it is linear.
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I believe the complexity of the simplex method is still an open research question; correct me if I'm wrong.
For most pivot rules (i.e., the rule to decide which variables enter and leave the basic solution), there exist counter examples of linear programs with exponential computation time. The question is whether a pivot rule exists that solves all linear programs in polynomial time.
The exponential time counter examples are often rather extreme. Small perturbations of the model parameters typically lead to problems that can be solved in polynomial time, hence it is quite rare to encounter an exponential time problem in practice.
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