OpenAI · 125-01 · 2026-09-24 · Theoretical computer science
Single-exponential recovery and bounded-price strictness for metric k-median
preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026
Family 125. The approximation threshold for metric -median.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Gives a deterministic polynomial-time -approximation for finite rational metric -median with specified candidate facilities, for every fixed . Assuming , the optimal infimum approximation factor is .
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 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 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 125: The metric <i>k</i>-median approximation threshold and recovery. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.