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Title: Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture
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.  more » « less
Award ID(s):
1900460
PAR ID:
10658022
Author(s) / Creator(s):
; ; ; ;
Editor(s):
Santhanam, Rahul
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
300
ISSN:
1868-8969
Page Range / eLocation ID:
14:1-14:48
Subject(s) / Keyword(s):
computational invariant theory geometric complexity theory orbit problems abc-conjecture closest vector problem Computing methodologies → Algebraic algorithms Computing methodologies → Combinatorial algorithms Theory of computation → Algebraic complexity theory
Format(s):
Medium: X Size: 48 pages; 1209084 bytes Other: application/pdf
Size(s):
48 pages 1209084 bytes
Sponsoring Org:
National Science Foundation
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