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

TL;DR

This paper explores the computational complexity of approximate orbit problems under torus actions, revealing NP-hardness for certain approximations and polynomial-time algorithms linked to the abc-conjecture, thus connecting complexity theory and number theory.

Contribution

It introduces the first complexity results for robust orbit problems, including NP-hardness proofs and algorithms contingent on the abc-conjecture, bridging invariant theory, lattice theory, and number theory.

Findings

NP-hardness for approximations with factor n^{Ω(1/ log log n)}

Polynomial-time algorithms for exponential approximation factors

Connection between algorithmic complexity and the abc-conjecture

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, B\"urgisser et al. 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 $\mathbb{C}^n$ up to a factor $\gamma>1$. 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 $\gamma = n^{\Omega(1/\log\log n)}$ 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 $\gamma = \exp(\mathrm{poly}(n))$. 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.