OpenAI claim summary for CS-142
computer sciencemathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 125; 2026-09-24).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Metric k-median at factor one plus two over e
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 125; 2026-09-24), the asserted result is that for every fixed positive $\varepsilon$, rational metric $k$-median has a deterministic $(1+2/e+\varepsilon)$ approximation that opens at most $k$ facilities.Mathematical statement of the claimed result
Manuscript 1.
For any fixed $\varepsilon>0$ and any finite rational metric $k$-median instance with facilities $F$, clients $J$, and $1\le k\le|F|$, the claimed deterministic algorithm returns $S\subseteq F$, $|S|\le k$, satisfying \[ \sum_{j\in J}d(j,S)\le(1+2/e+\varepsilon)\operatorname{OPT}_k. \] Running time is polynomial in full binary input length; its degree and constants may depend on the fixed accuracy $\varepsilon$.Outline of the argument
For manuscript 1 (family 125; 2026-09-24), a constant-surplus construction is made deterministic by compressing finite distributions and certifying scalar drift inequalities. Exact rational implementation controls rounding loss, and the budget-recovery reduction removes the surplus without losing the stated factor.Formal verification
For manuscript 1 (family 125; 2026-09-24), OpenAI lists Lean statements for this assertion: for every fixed positive $\varepsilon$, rational metric $k$-median has a deterministic $(1+2/e+\varepsilon)$ approximation that opens at most $k$ facilities.The selected statement includes the clause used for this problem.
The linked family Comparator checks are KMedianRecovery.lean, KMedianRefinedRecovery.lean, KMedianThreshold.lean, MetricKMedian.lean.
Their named declarations include OAI.MetricKMedian.main_finite_work, OAI.MetricKMedian.threshold, OAI.MetricKMedianRecovery.RefinedRecovery.refined_recovery, OAI.MetricKMedianRecovery.global_application.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, The approximation threshold for metric k-median, 2026-09-24; citation key
OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 125: Lean scope.
- Comparator KMedianRecovery.json.
- Comparator KMedianRecovery.lean.
- Comparator KMedianRefinedRecovery.json.
- Comparator KMedianRefinedRecovery.lean.
- Comparator KMedianThreshold.json.
- Comparator KMedianThreshold.lean.
- Comparator MetricKMedian.json.
- Comparator MetricKMedian.lean.
- OpenAI catalogue entry.