On the Convergence Time of a Natural Dynamics for Linear Programming

TL;DR

This paper analyzes a biologically inspired nonlinear differential equation system for solving linear programming problems, providing a new convergence time bound that highlights its efficiency and scalability.

Contribution

It introduces a novel convergence time bound for the Physarum dynamics in LPs, linking it to relative entropy and demonstrating its efficiency for positive-cost LPs.

Findings

Convergence time depends logarithmically on LP cost coefficients.

Bound scales linearly with inverse of desired accuracy.

Physarum dynamics efficiently approximates LP solutions.

Abstract

We consider a system of nonlinear ordinary differential equations for the solution of linear programming (LP) problems that was first proposed in the mathematical biology literature as a model for the foraging behavior of acellular slime mold Physarum polycephalum, and more recently considered as a method to solve LPs. We study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and derive a new time bound to approximate optimality that depends on the relative entropy between projected versions of the optimal point and of the initial point. The bound scales logarithmically with the LP cost coefficients and linearly with the inverse of the relative accuracy, establishing the efficiency of the dynamics for arbitrary LP instances with positive costs.