Existence of a polynomial-time pivot rule for the simplex method

Determine whether there exists a pivot rule for the simplex method that guarantees a polynomial-time worst-case running time (equivalently, a polynomial bound on the number of iterations) for all linear programs of the form maximize c^T x subject to Ax ≤ b.

Background

The paper discusses longstanding complexity questions surrounding the simplex method. While many specific pivot rules are known to admit exponential worst-case examples, it remains unknown whether there exists any pivot rule that ensures polynomial-time performance on all linear programs.

The authors highlight this as a central open question in linear optimization and relate it to other major open problems, emphasizing its historical significance dating back to the method’s inception.

References

The existence of an efficient pivot rule for the simplex method is a notorious open question since the inception of the method by Dantzig in 1947, and could yield a strongly polynomial algorithm for linear optimization, as well as a proof of the polynomial (monotone) Hirsch conjecture.

— An unconditional lower bound for the active-set method in convex quadratic maximization (2507.16648 - Bach et al., 22 Jul 2025) in Section 1 (Introduction)