On a Problem Posed by Steve Smale

TL;DR

This paper advances the understanding of Smale's 17th problem by developing a polynomial-time randomized algorithm for solving systems of complex polynomials, analyzing its smoothed and condition-based complexities, and proposing a nearly polynomial deterministic solution.

Contribution

It introduces a linear homotopy algorithm, extends the randomized LV algorithm, and provides a nearly polynomial deterministic algorithm for Smale's 17th problem.

Findings

The smoothed complexity of LV is polynomial in input size and inverse perturbation size.

The expected running time of LV depends on the condition of the input system.

A nearly polynomial average complexity deterministic algorithm is proposed for Smale's 17th problem.

Abstract

The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the average, in the size $N$ of the input system. A partial solution to this problem was given by Carlos Beltran and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV, a la Beltran-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and $\sigma^{-1}$, where $\sigma$ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system $f$, of the expected running time of LV with input $f$. In addition to its dependence on $N$ this bound also depends on the condition of $f$. Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is $N^{O(\log\log N)}$. This is nearly a solution to Smale's 17th problem.