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CS-143 · computer science · mathematics · Optimization

Metric k-median strictly below factor two

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is there one absolute $\sigma >0$ and a randomized polynomial-bit-time algorithm for arbitrary finite rational metric k-median that opens at most k facilities and achieves ratio $2-\sigma $ in expectation and with failure probability at most $(|clients|+|facilities|+2)^{-a}$ for any prescribed fixed $a>0$?

Short English statement

Is there one absolute $\sigma >0$ and a randomized polynomial-bit-time algorithm for arbitrary finite rational metric k-median that opens at most k facilities and achieves ratio $2-\sigma $ in expectation and with failure probability at most $(|clients|+|facilities|+2)^{-a}$ for any prescribed fixed $a>0$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-143-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-143

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 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

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