Natural algorithms for flow problems

In the last few years, there has been a significant interest in the computational abilities of Physarum polycephalum (a slime mold). This arose from a remarkable experiment which showed that this organism can compute shortest paths in a maze [10]. Subsequently, the workings of Physarum were mathematically modeled as a dynamical system and algorithms inspired by this model were proposed to solve several graph problems: shortest paths, flows, and linear programs to name a few. Indeed, computer scientists have initiated a rigorous study of these dynamics and a first step towards this was taken by [1,2] who proved that the Physarum dynamics for the shortest path problem are efficient (when edge-lengths are polynomially bounded). In this paper, we take this further: we prove that the discrete time Physarum-dynamics can also efficiently solve the uncapacitated min-cost flow problems on undirected and directed graphs; problems that are non-trivial generalizations of shortest path. This raises the tantalizing possibility that nature, via evolution, developed algorithms that efficiently solve some of the most complex computational problems, about a billion years before we did.