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Convergence of the Non-Uniform Physarum Dynamics

MPS-Authors
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Karrenbauer,  Andreas
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Kolev,  Pavel
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Mehlhorn,  Kurt
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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arXiv:1901.07231.pdf
(Preprint), 381KB

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Citation

Karrenbauer, A., Kolev, P., & Mehlhorn, K. (2019). Convergence of the Non-Uniform Physarum Dynamics. Retrieved from http://arxiv.org/abs/1901.07231.


Cite as: http://hdl.handle.net/21.11116/0000-0002-F39F-9
Abstract
Let $c \in \mathbb{Z}^m_{> 0}$, $A \in \mathbb{Z}^{n\times m}$, and $b \in \mathbb{Z}^n$. We show under fairly general conditions that the non-uniform Physarum dynamics \[ \dot{x}_e = a_e(x,t) \left(|q_e| - x_e\right) \] converges to the optimum solution $x^*$ of the weighted basis pursuit problem minimize $c^T x$ subject to $A f = b$ and $|f| \le x$. Here, $f$ and $x$ are $m$-vectors of real variables, $q$ minimizes the energy $\sum_e (c_e/x_e) q_e^2$ subject to the constraints $A q = b$ and $\mathrm{supp}(q) \subseteq \mathrm{supp}(x)$, and $a_e(x,t) > 0$ is the reactivity of edge $e$ to the difference $|q_e| - x_e$ at time $t$ and in state $x$. Previously convergence was only shown for the uniform case $a_e(x,t) = 1$ for all $e$, $x$, and $t$. We also show convergence for the dynamics \[ \dot{x}_e = x_e \cdot \left( g_e \left(\frac{|q_e|}{x_e}\right) - 1\right),\] where $g_e$ is an increasing differentiable function with $g_e(1) = 1$. Previously convergence was only shown for the special case of the shortest path problem on a graph consisting of two nodes connected by parallel edges.