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CS-142 · computer science · mathematics · Optimization

Metric k-median at factor one plus two over e

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed $\varepsilon >0$, is there a deterministic polynomial-bit-time algorithm for arbitrary finite rational metric k-median that opens at most k facilities and achieves approximation ratio $1+2/e+\varepsilon $?

Short English statement

For every fixed $\varepsilon >0$, is there a deterministic polynomial-bit-time algorithm for arbitrary finite rational metric k-median that opens at most k facilities and achieves approximation ratio $1+2/e+\varepsilon $?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-142-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-142

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).

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

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