The approximation threshold for metric $k$ -median: is $1+2/e$ attainable?

Theorem 1.1: for every fixed $\varepsilon>0$ , a deterministic algorithm running in time polynomial in the binary input length (the polynomial depends on $\varepsilon$ ) returns $S\subseteq F$ with $|S|\le k$ and cost at most $(1+2/e+\varepsilon)\mathrm{OPT}_k$ , for arbitrary $k$ and arbitrary finite rational metrics with specified candidate facilities. Corollary 1.2: under $P\ne NP$ , the infimum of polynomial-time approximation factors in this model is $1+2/e$ , the lower bound being the known coverage gap. It does not give a $(1+2/e)$ -approximation itself, does not settle the model where every client is a candidate facility ( $F=J$ ), whose lower bound differs, and claims no practical running time. The companion proves a randomized $(2-\sigma)$ -approximation for an absolute $\sigma>0$ by anchor recovery and bounded-price payment accounting.

Exact FrontierDelta

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The approximation threshold for metric $k$-median: is $1+2/e$ attainable? · Metric k-median threshold

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the question. lean/docs/125.md lists four challenges; two cover this entry: ComparatorChallenges/MetricKMedian.json (theorem OAI.MetricKMedian.main_finite_work, solution module OAI.Combinatorics.KMedian.Main) and KMedianThreshold.json (OAI.MetricKMedian.threshold, module OAI.Combinatorics.KMedian.Threshold). Both solution files exist at the pinned commit, and neither challenge is in the formalization catalogue formalization.yaml (only the companion's KMedianRecovery is). The statements were read here: for every $\varepsilon>0$ a polynomial-time Turing machine with finite alphabets outputs a nonempty set of at most $k$ candidate facilities of cost at most $(1+2/e+\varepsilon)\mathrm{OPT}$ on every finite rational instance; and, from a formalized $P\ne NP$ , the infimum of deterministic polynomial-time factors is exactly $1+2/e$ , so the hardness side is formalized too. That is the headline claim. Not rebuilt here. The threshold is an infimum: attaining $1+2/e$ itself is not claimed, and the $F=J$ model is a different question.