On the Complexity of Numerical Analysis

TLDR

The authors investigate two distinct frameworks for analyzing numerical complexity—the Blum–Shub–Smale real model and a generic floating‑point task—both centered on the PosSLP problem, and conjecture that transcendental constants add no computational power beyond nonuniform reductions to PosSLP. They establish that, in the BSS model with algebraic constants, polynomial‑time computation over the reals is equivalent to PosSLP, and that the generic floating‑point task is likewise equivalent to PosSLP, while also providing new insights into arithmetic‑circuit identity testing. Consequently, PosSLP is shown to lie in the counting hierarchy, implying that the Euclidean traveling salesman problem also resides in this class—improving the previous PSPACE upper bound—and that ACIT has subexponential complexity if factorials are not ultimately easy.

Abstract

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum–Shub–Smale model of computation over the reals; and (b) a problem we call the “generic task of numerical computation,” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether $N>0$. In the Blum–Shub–Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean traveling salesman problem lies in the counting hierarchy—the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.