A Universal Scaling Theory for Complexity of Analog Computation

TL;DR

This paper introduces a universal scaling theory for the complexity of solving large linear programming problems using analog computers modeled as dynamical systems, revealing universal behavior across different problem ensembles.

Contribution

It demonstrates that the complexity distributions of large linear programming problems follow universal scaling functions, independent of the specific probability ensemble.

Findings

Complexity distributions converge to universal scaling functions in large problem limits.

The scaling functions are analogous to those in phase transition theories.

Results extend previous findings from Gaussian ensembles to broader classes.

Abstract

We discuss the computational complexity of solving linear programming problems by means of an analog computer. The latter is modeled by a dynamical system which converges to the optimal vertex solution. We analyze various probability ensembles of linear programming problems. For each one of these we obtain numerically the probability distribution functions of certain quantities which measure the complexity. Remarkably, in the asymptotic limit of very large problems, each of these probability distribution functions reduces to a universal scaling function, depending on a single scaling variable and independent of the details of its parent probability ensemble. These functions are reminiscent of the scaling functions familiar in the theory of phase transitions. The results reported here extend analytical and numerical results obtained recently for the Gaussian ensemble.