Two results on slime mold computations

Abstract

We present two results on slime mold computations. In wet-lab experiments (Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems.Biologists proposed a mathematical model, a system of differential equations, for the slime's adaption process (J. Theoretical Biology'07). It was shown that the process convergences to the shortest path (J. Theoretical Biology'12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector.Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can ε-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on ε, and a number of steps with quartic dependence on opt/(εΦ), where Φ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost (opt).We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of ε, and the number of steps depends logarithmically on 1/ε and quadratically on opt/Φ.

Physarum Inspired Dynamics to Solve Semi-Definite Programs

“…C † (k0) is a constant matrix that will be defined in Section 3. Note that in case C ≻ 0 and the SDP is positive if we choose X as a full-rank matrix, then if the claim holds X(t) will converge to the optimum primal of (5). While seemingly strict, we will introduce an SDP which meets all the conditions and obtain a solution for the original SDP using it.…”

“…lim t→∞ c T x(t) = c T x * Numerous results have been derived by altering the dynamics. Some showcase using the discretized version of the dynamics in (4) and choose h such that the processing time of the dynamics improves, [5]. Others add a coefficient and create non-uniform dynamic of the form x i (t) = d i (q i (t) − x i (t)) [6].…”

A Laplacian approach to $$\ell _1$$-norm minimization

“…◻ Physarum dynamics. A previously studied dynamics that is formally similar to (4.2) is the Physarum dynamics [10,15,[43][44][45], namely, Differently from (4.2), the dynamics (4.9) is not a gradient flow, that is, there is no function f that allows to write the dynamics in the form (4.3) or (4.5) (with h the negative entropy). Still, from a qualitative point of view, the behavior of (4.9) appears to be rather similar to that of (4.2): namely, the trajectories still converge to an optimal solution of the associated (BP) problem (see, for example, [10, Theorem 2.9]).…”

“…The dynamics studied in Sections 4 and 5 bear some formal similarity to the so-called Physarum dynamics, studied in the context of natural computing, which are the network dynamics of a slime mold [10,15,[43][44][45]. The fact that Physarum dynamics are of IRLS type was first observed in [43].…”

“…In this context, our result can be seen as the derivation of a Physarum-like dynamics purely from an optimization principle: dissipation minimization following the natural gradient. A relevant difference is that the specific dynamics we study is a gradient system, while the dynamics studied by [10,43] is provably not a gradient system. This is precisely what enables us to apply the machinery of first-order convex optimization methods, and acceleration in particular.…”