As far as I understand, all know deterministic pivot rules for simplex algorithms have specific inputs on which the algorithm requires exponential time (or at least not polynomial) to find the optimum. Let us call these instances 'pathological' since usually (i.e. on most inputs) the simplex algorithm terminates quickly. I remember from my mathematical programming course that the standard examples of pathological instances for specific rules were highly structured. My general question is if this is an artifact of the specific examples, or a feature of pathological instances in general?
Results like smoothed analysis and the polynomial time algorithm extending it rely on perturbing the input --- suggesting that the pathological examples are very special. Hence the intuition that the pathological instances are highly-structured does not seem that far fetched.
Does anybody have any specific insights on this? Or some references to existing work? I have been specifically vague about what I mean by 'structured' to try to be as encompassing as possible, but suggestions on how to better pin down 'structured' would also be useful. Any advice or references are greatly appreciated!