2026-08-17T16:21:41Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20410 2024-07-15T12:26:45Z ddc:004 open_access

Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture Bürgisser, Peter Doğan, Mahmut Levent Makam, Visu Walter, Michael Wigderson, Avi computational invariant theory geometric complexity theory orbit problems abc-conjecture closest vector problem When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Peter Bürgisser and Mahmut Levent Doğan and Visu Makam and Michael Walter and Avi Wigderson 2024 Is Part Of LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2024.14 urn:nbn:de:0030-drops-204100 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14 eng https://creativecommons.org/licenses/by/4.0/legalcode