Approximation hardness of Euclidean closest vector
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How hard is it to approximate the lattice point nearest a target in Euclidean distance? The approximation factor is measured as a power of the lattice dimension n, and reduction and promise conventions are part of each hardness theorem.
Why it matters and what counts as progress
Why it matters: Polynomial-factor hardness marks a quantitative limit on approximation for a foundational lattice optimization problem.
What counts as progress: Prove NP-hardness for larger approximation factors through valid reductions, or improve approximation algorithms at comparable dimensions and promises.
How the problem developed
- 1981 · Hardness result [1]
Peter van Emde Boas Proved exact closest vector is NP-hard, establishing the decision problem's computational baseline.
- 2003 · Hardness result [2]
Irit Dinur, Guy Kindler, Ran Raz and Shmuel Safra Proved NP-hardness of approximating closest vector within almost-polynomial factors.
- 1 Aug 2026 · Hardness result [3]
OpenAI Reported a deterministic reduction proving NP-hardness for Euclidean GapCVP with gap n^(1/400).
- 4 Aug 2026 · Hardness result [4]
Zhao Song Strengthened the Euclidean gap hardness to n^(1/2−ε) for every fixed ε>0 under deterministic polynomial-time reductions.
AI contribution
Ten Advances in Mathematics and Theoretical Computer Science
A direct 3SAT reduction proves NP-hardness of approximating Euclidean closest vector within n^(1/400), under the dimension and promise conventions of Chapter 7. The chapter gives separate consequences for binary nearest codeword and finite p norms.
Sources
- Another NP-complete partition problem and the complexity of computing short vectors in a lattice
- Approximating CVP to Within Almost-Polynomial Factors Is NP-Hard
- OpenAI, Ten Advances in Mathematics, Chapter 7
- Hardness of Euclidean Closest Vector within n^(1/2−ε) and Binary Nearest Codeword within n^(1−ε)
- OpenAI, closest-vector formalization manifest
- Sienicki and coauthors, Chapter 7 human review
- Song, later closest-vector hardness result
- OpenAI, Ten Advances in Mathematics release and attribution
Website timeline
Publication dates and editorial changes.
Astra proof gives polynomial-factor hardness for closest vector