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Convergence of the Non-Uniform Directed Physarum Model

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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Fulltext (public)

arXiv:1906.07781.pdf
(Preprint), 687KB

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Citation

Facca, E., Karrenbauer, A., Kolev, P., & Mehlhorn, K. (2019). Convergence of the Non-Uniform Directed Physarum Model. Retrieved from http://arxiv.org/abs/1906.07781.


Cite as: http://hdl.handle.net/21.11116/0000-0005-1DBA-A
Abstract
The directed Physarum dynamics is known to solve positive linear programs: minimize $c^T x$ subject to $Ax = b$ and $x \ge 0$ for a positive cost vector $c$. The directed Physarum dynamics evolves a positive vector $x$ according to the dynamics $\dot{x} = q(x) - x$. Here $q(x)$ is the solution to $Af = b$ that minimizes the "energy" $\sum_i c_i f_i^2/x_i$. In this paper, we study the non-uniform directed dynamics $\dot{x} = D(q(x) - x)$, where $D$ is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with $D$ being the identity matrix), as it allows each component of $x$ to react with different speed to the differences between $q(x)$ and $x$. Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.