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

TL;DR

The paper investigates robust orbit-distance problems for torus actions, introducing the logarithmic distance $\delta_{\log}$ to linearize the action and avoid issues from non-closed orbits. It establishes NP-hardness for small approximation factors via a reduction from CVP and develops polynomial-time algorithms for larger factors under separation hypotheses linked to the abc-conjecture, supported by a novel lattice-lifting theorem that connects lattice projections to orbit distances. The work also integrates invariant theory and the Kempf-Ness framework to extend the approach and discusses conditional polynomial-time results, witness construction, and potential extensions to noncommutative settings. Overall, it reveals deep, surprising ties between computational complexity of orbit problems, lattice theory, and number theory, opening new avenues for both algorithm design and number-theoretic insights.

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. 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 $γ>1$. 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 $γ= \exp(\mathrm{poly}(n))$. 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.

Figures (2)

• Figure 1: (a) Action of $T = \mathbb{C}^\times$ on $V = \mathbb{C}^2$ given by $t\cdot (x,y)=(tx,ty)$: All orbits are lines through the origin, with the origin excluded. Thus, all orbit closures intersect in the origin. (b) Action of $T = \mathbb{C}^\times$ on $V = \mathbb{C}^2$ given by $t\cdot (x,y)=(tx,t^{-1}y)$: All orbits of vectors with non-zero coordinates $x,y\neq0$ are closed (they are hyperbolas).
• Figure 2: The logarithm maps the "hyperbolic" orbits (left) into parallel "lines" (right), which have a finite Euclidean distance. We define the logarithmic distance of the former as the Euclidean distance of the latter.

Theorems & Definitions (7)

• Theorem 4
• Theorem 5: Lattice lifting
• Theorem 8
• Theorem 9
• Theorem 10
• Theorem 12
• Theorem 13