Polynomial (monotone) Hirsch conjecture

Prove that the (monotone) graph diameter of every d-dimensional polytope with m facets is bounded above by a polynomial in d and m; equivalently, establish a polynomial upper bound on the length of the shortest (monotone) path between vertices along edges of the polytope.

Background

The classical Hirsch conjecture (linear bound) is known to be false, but the polynomial Hirsch conjecture—asserting a polynomial bound on polytope diameters—remains open, as does its monotone variant.

The authors connect progress on pivot rules and simplex complexity to potential advances on the (monotone) Hirsch conjecture, highlighting its status as a central conjecture in the field.

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. These are widely regarded as two of the most important open problems in the theory of linear programming.

— 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)

In fact, it is not even known whether—independent of any pivot rule—a path of polynomial length between any two extreme points of a polyhedron is guaranteed to exist. This is currently referred to as the polynomial Hirsch conjecture, and it is a major open question in discrete mathematics.

— Escaping Degeneracy by Following Shadow Edges  (2610.12130 - Black et al., 8 Oct 2026) in Section 1, Introduction