Strongly polynomial algorithm for linear programming

Determine whether there exists a strongly polynomial-time algorithm for linear programming that runs in time polynomial solely in the number of variables and constraints (independent of the numerical encoding length), for problems maximize c^T x subject to Ax ≤ b.

Background

Beyond weakly polynomial methods such as the ellipsoid and interior-point algorithms, it is unknown whether linear programming admits a strongly polynomial algorithm.

The authors note that a solution to the pivot-rule question for the simplex method could yield such an algorithm and explicitly identify this as one of the most important open problems in linear programming.

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)

By Proposition~\ref{prop:general-lp-hardness}, a strongly polynomial algorithm for the general model would resolve the longstanding open problem of strongly polynomial linear programming \citep[Problem~9]{Smale1998}.

— Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes  (2610.02131 - Zhong et al., 1 Oct 2026) in Section 3, immediately after Proposition 3.1 (Reduction from general linear programming)

Finding a strongly polynomial time algorithm for linear programming is a ‘holy grail’ open problem in discrete mathematics and theoretical computer science, listed indeed as one of the key mathematical problems for the next century by Smale.

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