Scaling and Universality of the Complexity of Analog Computation

TL;DR

This paper investigates the computational complexity of analog linear programming solvers, revealing universal scaling laws for complexity measures across different problem ensembles in large systems.

Contribution

It demonstrates the universality of the scaling functions for complexity measures in large analog computation problems, extending previous Gaussian ensemble results.

Findings

Probability distributions become universal scaling functions for large problems.

Scaling functions depend only on the variance of the problem ensemble.

Results support the conjecture of universality across different ensembles.

Abstract

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We analyze numerically various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.