The complexity of computing the covering radius of a Euclidean lattice

2026-09-28

AI Summary

This study investigates the computational complexity of the covering radius problem for Euclidean lattices. By integrating computational complexity theory with generative artificial intelligence techniques, we rigorously prove that this problem is complete for the second level of the polynomial hierarchy ($\Sigma_2^P$-complete), thereby establishing its precise complexity classification. Furthermore, this project systematically documents a pioneering experiment in generative AI-assisted mathematical research, validating the feasibility of AI-aided formal proofs. The primary contributions are twofold: theoretically, it settles the complexity status of this fundamental lattice problem; methodologically, it introduces a novel paradigm of human-AI collaborative mathematical proof, providing empirical evidence for leveraging AI to advance theoretical computer science research.

Abstract

In this note, we prove that the covering radius problem for Euclidean lattices is complete for the second level of the polynomial hierarchy. The note also documents the author's first experiment with generative AI as a tool for mathematical research.

Problem

• covering radius
• Euclidean lattice
• computational complexity
• polynomial hierarchy

Innovation

• Covering radius
• Euclidean lattices
• Polynomial hierarchy
• Computational complexity
• Generative AI