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

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

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Peter Bürgisser
  • Institute of Mathematics, Technische Universität Berlin, Germany
Mahmut Levent Doğan
  • Institute of Mathematics, Technische Universität Berlin, Germany
Visu Makam
  • Radix Trading, Amsterdam, The Netherlands
Michael Walter
  • Faculty of Computer Science, Ruhr-Universität Bochum, Germany
Avi Wigderson
  • School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA

Funding

  • Bürgisser, Peter: Supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 787840).
  • Doğan, Mahmut Levent: Supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 787840).
  • Makam, Visu: Partially supported by NSF Grant CCF-1900460 and the University of Melbourne.
  • Walter, Michael: Supported by the European Union (ERC, SYMOPTIC, 101040907), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972, by the BMBF (QuBRA, 13N16135), and by the Dutch Research Council (NWO grant OCENW.KLEIN.267).
  • Wigderson, Avi: Supported by the NSF Grant CCF-1900460.

Acknowledgements

The authors thank Matías Bender, Alperen A. Ergür, Jonathan Leake, and Philipp Reichenbach for productive discussions.

Cite AsGet BibTex

Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.14

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, 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.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Computing methodologies → Combinatorial algorithms
  • Theory of computation → Algebraic complexity theory
Keywords
  • computational invariant theory
  • geometric complexity theory
  • orbit problems
  • abc-conjecture
  • closest vector problem

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