OpenAI claim summary for CS-143
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 strictly below factor two
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 metric $k$-median admits exact-budget recovery from nearly complete proxy anchors, and a compatible bounded-price argument yields an absolute factor below two on arbitrary rational metrics.Mathematical statement of the claimed result
Manuscript 1.
For finite rational metric $k$-median with clients $D$, facilities $F$, and $1\le k\le|F|$, define $\operatorname{cost}(S)=\sum_{j\in D}d(j,S)$ and $\operatorname{opt}_k$ as its minimum over $|S|\le k$. The claim gives an absolute $\sigma>0$ such that for every fixed $a>0$ a randomized polynomial-bit-time algorithm always returns a feasible $S$ and satisfies \[ \Pr[\operatorname{cost}(S)\le(2-\sigma)\operatorname{opt}_k] \ge1-(|D|+|F|+2)^{-a}. \] It can additionally achieve $\mathbb E\operatorname{cost}(S)\le(2-\sigma)\operatorname{opt}_k$. Polynomial constants may depend on fixed parameters.Outline of the argument
For manuscript 1 (family 125; 2026-09-24), sampled balls are completed using a partition-constrained submodular maximization that separates destinations without sampled proxies. A stability dichotomy combines recovery with one certified surplus execution, paying for removals while opening no more than the original facility budget.Formal verification
For manuscript 1 (family 125; 2026-09-24), OpenAI lists Lean statements for this assertion: metric $k$-median admits exact-budget recovery from nearly complete proxy anchors, and a compatible bounded-price argument yields an absolute factor below two on arbitrary rational metrics.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, Single-exponential recovery and bounded-price strictness for metric k-median, 2026-09-24; citation key
OAI:Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: main:theorem. - 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.