A strongly polynomial algorithm for linear programming

Linear programs can be solved in time polynomial in the number of bits of their input, but every known polynomial algorithm slows down when the numbers get longer. Is there an algorithm whose number of arithmetic steps depends only on the numbers of variables and constraints? This is Smale's ninth problem for the 21st century.

Problem

A linear program asks to maximize $c^\top x$ subject to $Ax\le b$, for given $A\in\mathbb{Q}^{m\times n}$, $b\in\mathbb{Q}^m$ and $c\in\mathbb{Q}^n$. An algorithm for it is strongly polynomial if

• the number of arithmetic operations it performs (additions, subtractions, multiplications, divisions and comparisons) is bounded by a polynomial in $m$ and $n$ alone, whatever the sizes of the entries, and
• every number it computes has bit length bounded by a polynomial in the bit length of the input.

Question. Is there a strongly polynomial algorithm for linear programming? Smale's ninth problem asks the same for deciding whether a system $Ax\ge b$ is feasible, in the real-number model of computation [2].

Background

Khachiyan's ellipsoid method and Karmarkar's interior-point method (1984) solve linear programs in time polynomial in the input's bit length, so they are only weakly polynomial: their step counts grow with the size of the numbers. Megiddo raised the question of a genuinely polynomial algorithm in 1982–83 [1], and Smale listed it in 1998 among his problems for the next century [2]. It is Problem 8 of The Open Problems Project, in its original list of thirty.

What is known

• Fixed dimension. For any fixed number of variables, linear programming takes time linear in the number of constraints (Megiddo, 1984, and Dyer, 1984, for two and three variables) [3]. In general dimension, a randomized algorithm of Matoušek, Sharir and Welzl (1996) runs in subexponential time.
• Special classes. Tardos (1986) gave an algorithm whose number of operations depends only on the size of the entries of $A$, not of $b$ and $c$; this covers network flow problems [4]. Dadush, Koh, Natura, Olver and Végh (2024) gave a strongly polynomial algorithm for linear programs with at most two nonzero entries in every row or in every column of $A$ [6].
• Barriers. Primal-dual log-barrier interior-point methods are not strongly polynomial: Allamigeon, Benchimol, Gaubert and Joswig (2018) built linear programs with $3r+1$ constraints in dimension $2r$ on which they need $\Omega(2^r)$ iterations [5]. Allamigeon, Gaubert and Vandame (STOC 2022) extended this to every interior-point method based on a self-concordant barrier. A strongly polynomial simplex method would need polynomially many pivots on every input, which would imply the polynomial Hirsch conjecture, an open problem on the diameters of polytopes. For the circuit diameter, a relaxation that allows steps along circuit directions, Natura (2026, unrefereed preprint) proved a strongly polynomial bound $O(m^2\log m)$ for polyhedra $\{x: Ax=b,\ x\ge 0\}$ with $m$ equality constraints; a circuit-augmentation algorithm achieving it would solve the problem.
• Unaccepted claims. Several arXiv preprints claim strongly polynomial algorithms for general linear programs, for example arXiv:2006.11466 (2020) and arXiv:2503.12041 (2025, revised 2026). None has been accepted, and recent work by specialists still treats the problem as open.

What would count as progress

• A strongly polynomial algorithm for a broader class of linear programs, for example with at most three nonzero entries per column, with a proof of the operation count.
• Exponential lower bounds for further families of methods, such as path-following schemes that are not based on self-concordant barriers, in the style of [5].
• A strongly polynomial algorithm for linear programming in general.

References

1. N. Megiddo, Towards a genuinely polynomial algorithm for linear programming, SIAM J. Comput. 12 (1983), 347–353. doi:10.1137/0212022
2. S. Smale, Mathematical problems for the next century, Math. Intelligencer 20(2) (1998), 7–15. doi:10.1007/BF03025291
3. N. Megiddo, Linear programming in linear time when the dimension is fixed, J. ACM 31 (1984), 114–127. doi:10.1145/2422.322418
4. É. Tardos, A strongly polynomial algorithm to solve combinatorial linear programs, Oper. Res. 34 (1986), 250–256. doi:10.1287/opre.34.2.250
5. X. Allamigeon, P. Benchimol, S. Gaubert and M. Joswig, Log-barrier interior point methods are not strongly polynomial, SIAM J. Appl. Algebra Geom. 2 (2018), 140–178. doi:10.1137/17M1142132, arXiv:1708.01544
6. D. Dadush, Z. K. Koh, B. Natura, N. Olver and L. A. Végh, A strongly polynomial algorithm for linear programs with at most two nonzero entries per row or column, Proc. STOC 2024, 1561–1572. doi:10.1145/3618260.3649764

Source

Imported from The Open Problems Project, Problem 8, curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.