Polynomial worst-case complexity of the simplex method

Determine whether there exists a pivot rule for the simplex method that guarantees a polynomial upper bound on the number of pivots for all linear programs in standard form (minimize c^T x subject to Ax = b and x ≥ 0).

Background

The simplex method typically performs very well in practice, yet worst-case constructions (e.g., Klee–Minty) show exponential behavior for classical pivot rules. Despite extensive paper of many deterministic and randomized pivot rules, no general polynomial worst-case bound is known for all LP instances.

Average-case analyses and, more recently, smoothed analysis have provided explanations for why simplex performs well on typical or slightly perturbed instances, but these do not resolve the central worst-case question for an explicit pivot rule on arbitrary inputs.