Deciding whether a lattice has an orthonormal basis is in co-NP

• Published in: Mathematical programming Vol. 208; no. 1-2; pp. 763 - 775
• Main Author: Hunkenschröder, Christoph
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2024
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Short Communication; Theoretical; Co-NP; Self-dual lattice; Lattice isomorphism; Characteristic vectors
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-023-02052-1

We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n , i.e. a rational matrix B ∈ Q 4 n × n such that B ⊺ B = G .