Natural Algorithms for Flow Problems

2016

J. Math. Biol.

Optimization of fluid transport in the slime mold Physarum polycephalum has been the subject of several modeling efforts in recent literature. Existing models assume that the tube adaptation mechanism in P. polycephalum's tubular network is controlled by the sheer amount of fluid flow through the tubes. We put forward the hypothesis that the controlling variable may instead be the flow's pressure gradient along the tube. We carry out the stability analysis of such a revised mathematical model for a paralleledge network, proving that the revised model supports the global flow-optimizing behavior of the slime mold for a substantially wider class of response functions compared to previous models. Simulations also suggest that the same conclusion may be valid for arbitrary network topologies.

Section: Introductionmentioning

confidence: 99%

A revised model of fluid transport optimization in Physarum polycephalum

2016

J. Math. Biol.

“…Let Note that the first term is nonnegative whenever x(t) is feasible for the LP, and the second term is always nonnegative. Similar to previous analysis of Physarum dynamics based on potential functions [4,17,18], the potential function Φ contains a term that depends on the cost of the candidate solution x, and an "entropic barrier" term that captures the geometry of the feasible region: in particular, the second term penalizes distributions that get too close to the boundary of the positive orthant whenever the corresponding coordinate of the optimal solution is not on the boundary (that is, ξ j (t) ≈ 0 but ξ * j > 0). A difference with respect to previous papers is that the potential function (7) is dimensionless, which is natural since our aim is to bound the relative, rather than absolute, approximation error.…”

Section: Convergence In Cost Valuementioning

confidence: 69%

On the Convergence Time of a Natural Dynamics for Linear Programming

2019

Algorithmica

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. References 11

“…The convergence proof gives an upper bound on the step size and on the number of steps required until an ε-approximation of the optimum is obtained. [SV16b] extends the result to the transshipment problem and [SV16a] further generalizes the result to the case of positive LPs. The paper [SV16b] is related to our first result.…”

Section: The Biologically-inspired Modelmentioning

confidence: 77%

Two results on slime mold computations

Becker

Theoretical Computer Science

Karrenbauer

et al. 2019

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.