Result 125, Theoretical computer science

The metric k-median approximation threshold and recovery

Gives a deterministic polynomial-time $(1+2/e+\varepsilon)$ -approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming $P\ne NP$ , the optimal infimum approximation factor is $1+2/e$ .

Lean formalization New or sharp bound

The bigger picture

Why it matters

How close can efficient algorithms come to the best placement of a limited number of facilities? The manuscript reports a sharp approximation threshold, identifying both an achievable guarantee and a conditional barrier to doing better.

What changes?

In metric k-median, one chooses at most k sites from specified candidate facilities to minimize the sum of clients' distances to their nearest chosen site. Distances form a finite metric: they are symmetric, obey the triangle inequality, and here are rational numbers. The threshold manuscript reports a deterministic polynomial-time algorithm for every fixed positive epsilon, with cost at most 1 + 2/e + epsilon times the optimum, where e is the base of natural logarithms.

What does that help mathematicians do?

Assuming P is not equal to NP, the reported guarantee matches the lower limit on polynomial-time approximation factors: 1 + 2/e, about 1.736. Researchers can therefore rule out any uniformly smaller factor in this model without overturning that assumption. The word "infimum" matters: guarantees can approach the boundary arbitrarily closely, but the claim does not assert an algorithm attaining it exactly.

Are there practical applications?

The practical connection is facility placement: choosing a limited set of service sites while controlling total distance to clients. The demonstrated contribution is a worst-case guarantee for the abstract metric model, not a deployment result. Polynomial time alone does not establish practical speed, especially when a very small epsilon is requested.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Single-exponential recovery and bounded-price strictness for metric k-median

September 24, 2026 53 pages Main result formalized in Lean

We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a $(1+2/e+\varepsilon)$ approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction. Together the recovery and payment arguments give a randomized $(2-\sigma)$ approximation, for an absolute σ > 0, on arbitrary finite rational metrics, both with high probability and in expectation, while opening at most k facilities on every output.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026,
  author = {{OpenAI}},
  title = {{Single-exponential recovery and bounded-price strictness for metric $k$-median}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf}{OAI:Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026}},
  year = {2026}
}

The approximation threshold for metric k-median

September 24, 2026 46 pages

For every fixed ε > 0, we give a deterministic polynomial-time $(1+2/e+\varepsilon)$ -approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under $P\ne NP$ , the infimum approximation factor in this model is therefore $1+2/e$ .

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026,
  author = {{OpenAI}},
  title = {{The approximation threshold for metric $k$-median}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf}{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/125.md.

The metric k-median approximation threshold and recovery

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives an exact-budget recovery algorithm for metric $k$ -median on the stated polynomially bounded integral metrics. Suppose a supplied anchor represents all but logarithmically many comparison clusters by distinct proxies, with total proxy cost at most the comparison cost plus a sufficiently small relative error. The algorithm always opens at most $k$ facilities and, with any requested confidence, attains a $(1+2/e+\varepsilon)$ factor relative to those comparison centers in polynomial time.

The linked final approximation result applies to arbitrary finite rational metrics. For one absolute $\sigma>0$ , a polynomial-time fair-bit algorithm always returns a feasible solution, has expected cost at most $(2-\sigma)\mathrm{OPT}$ , and attains the same factor with arbitrarily high polynomial confidence.

The formalization gives, for every fixed $\varepsilon>0$ , a deterministic polynomial-time $(1+2/e+\varepsilon)$ -approximation for metric $k$ -median on finite rational metrics with specified candidate facilities. Every output is a nonempty subset of the candidate facilities and opens at most $k$ of them.

Assuming $P\ne NP$ , it also proves that the infimum of all polynomial-time approximation factors in this model is exactly $1+2/e$ .

Comparator links

Result	Comparator statement
Randomized metric $k$ -median approximation below two	KMedianRecovery.lean
Exact-budget recovery from accurate distinct cluster proxies	KMedianRefinedRecovery.lean
Exact approximation threshold under $P\ne NP$	KMedianThreshold.lean
Deterministic metric $k$ -median approximation at the threshold	MetricKMedian.lean

At a glance

Subject: Theoretical computer science
Manuscripts: 2
Pages: 99
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

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