Noise resilience of deterministic analog combinatorial optimization solvers

TL;DR

This paper investigates how noise affects the performance of analog dynamical systems designed to solve NP-hard combinatorial problems, revealing a threshold behavior and scaling limits for solution accuracy.

Contribution

It provides a numerical analysis of noise impact on two analog solvers for Ising and k-SAT problems, identifying solution robustness thresholds and their scaling with problem size.

Findings

Solution-finding ability remains stable below a noise threshold

Solution quality deteriorates rapidly above the noise threshold

Noise tolerance thresholds decrease algebraically with problem size

Abstract

Several continuous dynamical systems have recently been proposed as special-purpose analog computers designed to solve combinatorial optimization problems such as $k$-SAT or the Ising problem. While combinatorial optimization problems are known to be NP-hard, and thus scale, in the worst case, exponentially with the problem size, these analog solvers promise substantial speed-up and scaling advantages in finding the solution. The underlying algorithms, which can be cast in the form of differential equations, generically involve highly chaotic dynamics and thus assume that the system variables can be processed with, in principle, arbitrary precision. However, both actual physical systems as well as finite digital machines, which are used to virtually emulate the dynamics, can process the evolution only with finite precision, be it because of intrinsic noise or because of limited precision in number representation. We investigate the impact of such noise on the solution-finding capability. To this end, we focus on two representative analog solvers, designed to address the Ising problem and the $k$-SAT problem, respectively. Our numerical analysis reveals that the ability of these algorithms to find solutions exhibits a threshold behavior under the addition of noise, where the solution-finding capability remains mostly uncompromised below a noise threshold, while it rapidly deteriorates above the threshold. As we show, these noise tolerance thresholds decrease with the problem size, following an approximate algebraic scaling. This allows us to infer principal limits on the problem sizes that can be efficiently tackled with these solvers under given noise levels.