Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

Peter Bürgisser*, Mahmut Levent Doğan*, Visu Makam*, Michael Walter*, Avi Wigderson*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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, [16] 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 = nΩ(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.

Original languageEnglish
Title of host publication39th Computational Complexity Conference, CCC 2024
EditorsRahul Santhanam
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773317
DOIs
StatePublished - Jul 2024
Externally publishedYes
Event39th Computational Complexity Conference, CCC 2024 - Ann Arbor, United States
Duration: 22 Jul 202425 Jul 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume300
ISSN (Print)1868-8969

Conference

Conference39th Computational Complexity Conference, CCC 2024
Country/TerritoryUnited States
CityAnn Arbor
Period22/07/2425/07/24

Bibliographical note

Publisher Copyright:
© Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson.

Keywords

  • abc-conjecture
  • closest vector problem
  • computational invariant theory
  • geometric complexity theory
  • orbit problems

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