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CS-85 · computer science · mathematics · Algorithms and data structures · Optimization

Smale's 9th Problem: Linear Programming in Polynomial Time

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Statement

Definitions and mathematical statement

Shared notation.
Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Consider rational linear programmes $\min\{c^Tx:Ax\leq b\}$ with $n$ variables and $m$ inequalities. Strong polynomiality means that the number of arithmetic operations and comparisons is bounded by a polynomial in $m+n$, independently of the coefficient bit lengths, while the intermediate rational numbers have encoding lengths polynomial in the input encoding length. The algorithm must also correctly detect infeasibility or an unbounded objective. [S1] [S2]

Statement recorded in the supplied source.
Find a strongly polynomial algorithm for linear programming. [S1]

Short English statement

Can every rational linear programme be solved with an arithmetic-operation count depending polynomially only on its numbers of variables and constraints?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 2

CS-85-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-85

Strongly Polynomial Algorithms for Linear Programming

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A strongly polynomial algorithm uses a number of arithmetic operations polynomial in the numbers of variables and constraints, independent of input coefficient bit lengths, while the bit lengths of intermediate values remain polynomially bounded in the input length.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. Source-record limitation: Exact duplicate of the existing strongly polynomial linear-programming target under canonical ID 52.

Related catalogue records: 52. Separate source IDs are preserved for their provenance.

Statement recorded in the supplied source.
Does linear programming admit a strongly polynomial-time algorithm? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Solve every linear program with a strongly polynomial algorithm.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006101 and code OWR-14298804-001 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 52. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

Attempted route: remove geometric conditioning by normalizing rows

The report concerns progress for two-variable-per-inequality LPs, while this record repeats the unrestricted strongly polynomial question. A tempting extension of geometry-dependent pivot bounds is to normalize every constraint normal and hope that the resulting angular parameter is bounded using only the number of rows and columns. For $B\ge1$, take two rational normals \[ a=(1,0),\qquad b=(1,2^{-B}). \] Their unit normalizations have distance-to-span parameter \[ \delta_B=\operatorname{dist}\left(b/\|b\|_2,\operatorname{span}(a)\right) =\frac{2^{-B}}{\sqrt{1+2^{-2B}}}. \]
Proposition: Positive rescaling of individual rows cannot give a positive lower bound on this parameter depending only on the two fixed dimensions.
Proof: The span of $a$ is the horizontal axis, so the distance of normalized $b$ is the absolute value of its second coordinate, giving the displayed formula. For positive $s,t$, normalizing $sa,tb$ gives exactly the same two unit vectors. Their parameter tends to zero as $B$ increases, although there are always two rows and two columns. The unnormalized coefficients have rational encodings of length $O(B)$, so this is a legitimate family of finite inputs. ∎
Consequently a proved operation bound containing $\delta_B^{-1}$ cannot be turned into a strongly polynomial bound merely by this row normalization: its displayed bound grows at least $2^B$. This is an obstruction to that deduction, not a lower bound on the actual execution time of every geometric algorithm. Arbitrary column transformations and different pivot rules are not excluded.

Verification and remaining gaps

Exact arithmetic gives the unnormalized determinants $2^{-B}$ at $B=1,10,100$; normalized distances were also evaluated. No algorithm with operation count polynomial only in the row and column counts is obtained. Restricted LP advances in the original report are not promoted to a solution of general LP.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 pp. 2950--2951 and 2965--2966, restricted 2VPI progress. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.
Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch011/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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CS-85-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-85

Smale's Strongly Polynomial Linear Programming Problem

Definitions and mathematical statement

Shared notation.
Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Consider rational linear programmes $\min\{c^Tx:Ax\leq b\}$ with $n$ variables and $m$ inequalities. Strong polynomiality means that the number of arithmetic operations and comparisons is bounded by a polynomial in $m+n$, independently of the coefficient bit lengths, while the intermediate rational numbers have encoding lengths polynomial in the input encoding length. The algorithm must also correctly detect infeasibility or an unbounded objective. [S1] [S2]

Statement recorded in the supplied source.
Find a strongly polynomial algorithm for linear programming. [S1]

Short English statement

Can every rational linear programme be solved with an arithmetic-operation count depending polynomially only on its numbers of variables and constraints?

Research attempt

Status: UNRESOLVED. We give a strongly polynomial algorithm for the one-variable case, extend it to systems with independent coordinates, and identify the coefficient-independent counting obstruction in vertex enumeration and elimination. The argument does not solve general linear programming.
Formulation and source check.
Smale's original problem list [E1], consulted 1 October 2026, asks for a polynomial algorithm over the reals deciding linear-inequality feasibility. The canonical definition here additionally specifies rational inputs, intermediate bit lengths, optimization, and detection of unboundedness. All claims below use that explicit definition. A running time polynomial in the total input bit length alone does not answer it. Conversely, arithmetic-operation counts alone are insufficient if the rational numbers created have uncontrolled encodings.
A complete algorithm in one variable.
Write the constraints as $a_ix\le b_i$, with rational $a_i,b_i$. If $a_i=0$, reject when $b_i<0$ and otherwise discard the row. If $a_i>0$, it gives an upper bound $x\le b_i/a_i$; if $a_i<0$, it gives a lower bound $x\ge b_i/a_i$. Let \[ L=\max_{a_i<0} b_i/a_i,\qquad U=\min_{a_i>0} b_i/a_i, \] where a missing lower bound is $-\infty$ and a missing upper bound is $+\infty$. The feasible set is exactly $[L,U]$, interpreted with the relevant missing endpoints. It is empty precisely when $L>U$ or a discarded zero row was inconsistent.

For objective $cx$, if $c>0$ the minimum is $cL$ attained at $L$ when $L$ is finite, and is unbounded below otherwise. If $c<0$, use $U$ in the same way. If $c=0$, every feasible point is optimal: choose zero if allowed, otherwise a finite endpoint. For example, an interval with no finite lower endpoint always contains either zero or its finite upper endpoint, so this selection covers one-sided and fully unbounded feasible sets.

The algorithm uses one pass, $O(m)$ arithmetic operations and comparisons, independent of numerical magnitudes. Rational division and comparison use only products and quotients of pairs of input numerators and denominators. Each retained endpoint is an original ratio, so its bit length is at most a constant times the total input bit length. Cross multiplication for comparisons has the same bound. Thus both parts of the strong-polynomial definition are satisfied for $n=1$.

Independent-coordinate systems.
Suppose each row of $A$ has at most one nonzero entry. Group the inequalities by their nonzero coordinate; zero rows are treated as above. Feasibility factors into a product of intervals. Optimize $c_jx_j$ independently on each interval. If one interval is empty, the whole system is infeasible. If a feasible interval is unbounded in an improving objective direction, the full problem is unbounded below: fix feasible values in all other coordinates and move only that coordinate. Otherwise the choices above yield an optimal vector.

This uses $O(m+n)$ arithmetic operations after identifying the nonzero entries, or $O(mn)$ when reading a dense input matrix. Both counts are polynomial in $m+n$. The bit-length argument remains coordinatewise and introduces no iterative coefficient growth. This is a complete special-class theorem, not a claim about sparse matrices in general: a row involving two coordinates already couples their feasible intervals.

Why vertex enumeration does not finish the problem.
For a bounded full-dimensional feasible polytope, a linear objective has an optimal vertex. Enumerating $n$-element subsets of the $m$ constraints, solving each full-rank equality system, and testing feasibility will therefore find an optimum in this restricted case. Even granting all linear-algebra and bit-length details, the number of subsets is $\binom mn$. For $m=2n$, \[ \binom{2n}{n}\ge\frac{2^{2n}}{2n+1}, \] because the middle binomial coefficient is the largest among $2n+1$ coefficients summing to $2^{2n}$. The enumeration count is exponential in $n$. Its independence from coefficient size does not make it polynomial in dimension.

The restriction to bounded full-dimensional polytopes is also substantive. A nonempty affine line has no vertices, while a constant objective on it has a finite attained optimum. An algorithm that reports infeasibility merely because it found no basic feasible point would fail on such an input. Adding an arbitrary large box requires a justified size bound and cannot silently repair this issue.

Fourier--Motzkin elimination: a second failed route.
To eliminate one variable, solve every row with a positive coefficient for an upper bound on that variable and every row with a negative coefficient for a lower bound. Pairing each lower bound with each upper bound enforces the necessary and sufficient compatibility inequalities in the remaining variables. With $p$ lower and $q$ upper bounds, this step can create $pq$ rows, in addition to zero-coefficient rows. Sufficiency follows because any assignment satisfying every pair has maximum lower bound at most minimum upper bound, so an eliminated variable can be selected between them.

This is an exact feasibility reduction, but it is not a polynomial-size guarantee. A roughly balanced split can square the number of rows at a step; repeated elimination yields a rapidly growing a priori bound. This observation is not a lower bound for every conceivable elimination algorithm, because redundancy removal could discard many rows. The missing theorem would have to show that enough rows can always be discarded efficiently while preserving the projected feasible set. We have not proved such a theorem.

Adversarial verification and remaining gap.
Zero coefficients, inconsistent constant rows, both signs of the objective, infinite endpoints, and empty intervals were explicitly handled in the one-variable proof. A chain of constraints may have extremely small or large rational endpoints without changing its arithmetic-operation count; exact rational comparisons prevent numerical tolerance errors. These checks do not transfer separability to a coupled system.

The first unresolved step is a dimension-polynomial bound on the work needed for arbitrary coupled inequalities, while maintaining polynomial intermediate bit lengths and valid infeasibility or unboundedness conclusions. The two failed routes isolate different obstructions: exponential numbers of bases and uncontrolled numbers of projected constraints. No numerical experiment, new general complexity lower bound, or solution of Smale's problem is claimed.

Sources

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