On a Natural Dynamics for Linear Programming

TL;DR

This paper interprets Physarum-inspired dynamics as a Riemannian steepest descent method for linear programming, proving global convergence to optimal solutions and demonstrating efficient discretization for certain LP classes.

Contribution

It provides a novel optimization interpretation of Physarum dynamics as a convex program with entropy regularization, establishing convergence and discretization properties.

Findings

Physarum dynamics are paths of convex optimization with entropy barrier.

Solutions of Physarum dynamics converge to LP optima.

Discretized Physarum dynamics are efficient for specific LPs.

Abstract

In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs [NTY00, IJNT11, JZ12]. These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming.