Existence of a strongly polynomial-time algorithm for linear programming

Determine whether a strongly polynomial-time algorithm exists for general linear programming, i.e., an algorithm whose running time is polynomial in the number of variables and constraints and independent of numerical data bit-lengths.

Background

The paper observes that finding a consistent linear threshold function can be phrased as a linear programming feasibility problem. In discussing computational subtleties, the authors note a classic unresolved question in optimization: whether one can solve linear programs in time that is polynomial in the combinatorial dimensions only, independent of input bit-length (strongly polynomial time).

This question, while not specific to PAC learning, influences the theoretical efficiency guarantees of LP-based learning methods and is a foundational open problem in algorithmic optimization.

References

Another is that the existence of a ``strongly polynomial time'' algorithm for linear programing is a famous open question; we refer the interested reader to for a discussion of these and related issues.

— The Probably Approximately Correct Learning Model in Computational Learning Theory (2511.08791 - Servedio, 11 Nov 2025) in Footnote in Section 4.1 (Distribution-free PAC learning from random examples: Linear programming and learning linear threshold functions)