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.

翻译：当群作用于集合时，它自然地将其划分为轨道，从而产生轨道问题。这类问题是自然的算法问题，因为对称性在物理学、数学、计算机科学、优化等众多领域的问题与结构中居于核心地位。因此，理解其计算复杂性具有重要意义。最近，Bürgisser等人首次给出了环面作用（即交换连续群在欧几里得空间上的作用）轨道问题的多项式时间算法。在本工作中，受理论与实际应用的驱动，我们研究了这些轨道问题的鲁棒推广形式的计算复杂性，其核心在于以因子γ>1近似计算ℂⁿ中轨道之间的距离。特别地，这使得我们能够判定两个输入是否近似处于同一轨道，或者相距甚远。一方面，我们通过将格的最短向量问题归约至此，证明了该问题在γ=n^{Ω(1/log log n)}时是NP难的。另一方面，我们描述了在近似因子γ=exp(poly(n))下求解该问题的算法。我们的算法结合了不变量理论与算法格论的工具，并且能够提供证明给定轨道邻近性的群元素（与先前工作的代数算法不同）。我们证明，这些算法在多项式时间内运行当且仅当著名的数论abc猜想的一个版本成立——这便在计算复杂性与数论之间建立了新颖而令人惊讶的联系。