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 language | English |
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Title of host publication | 39th Computational Complexity Conference, CCC 2024 |
Editors | Rahul Santhanam |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
ISBN (Electronic) | 9783959773317 |
DOIs | |
State | Published - Jul 2024 |
Externally published | Yes |
Event | 39th Computational Complexity Conference, CCC 2024 - Ann Arbor, United States Duration: 22 Jul 2024 → 25 Jul 2024 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 300 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 39th Computational Complexity Conference, CCC 2024 |
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Country/Territory | United States |
City | Ann Arbor |
Period | 22/07/24 → 25/07/24 |
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