Approximation threshold for metric k-median
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For finite rational metric k-median with specified candidate facilities, seek a deterministic polynomial-time (1+2/e+ε)-approximation opening at most k facilities for every fixed ε>0.
Why it matters and what counts as progress
Why it matters: It asks whether the algorithmic factor can meet the known P≠NP hardness threshold in infimum while retaining the exact facility budget.
What counts as progress: A polynomial-time factor approaching 1+2/e while opening at most k facilities; extra-facility solutions do not meet this target.
How the problem developed
- 2002 · Hardness result [1]
Kamal Jain, Mohammad Mahdian and Amin Saberi Proved that approximating metric k-median below 1+2/e is NP-hard, using a reduction from maximum coverage.
- 14 Mar 2025 · Algorithm [2]
Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee, Chris Schwiegelshohn and Ola Svensson Gave a (2+ε)-approximation, improving the previous best factor of 2.613.
- 24 Sept 2026 · Reported claim [3]
OpenAI The preprint claims a deterministic (1+2/e+ε)-approximation for finite rational metrics with a specified candidate-facility set, matching the hardness threshold as an infimum if P≠NP.
AI contribution
The approximation threshold for metric k-median
For every fixed ε>0, a deterministic polynomial-time algorithm on finite rational metric k-median instances with specified candidate facilities opens at most k facilities and attains factor 1+2/e+ε. Together with known P≠NP hardness, this gives infimum approximation factor 1+2/e.
Sources
- A new greedy approach for facility location problems
- A (2+ε)-Approximation Algorithm for Metric k-Median
- OpenAI, Metric k-median theorem and prior bounds
- The approximation threshold for metric k-median
- OpenAI, Metric k-median formalization scope
- OpenAI, Lean metric k-median algorithm proof
- OpenAI, Lean k-median threshold proof
- OpenAI mathematics release overview
Website timeline
Publication dates and editorial changes.