Two Results on Slime Mold Computations

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 $\varepsilon$-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on $\varepsilon$, and a number of steps with quartic dependence on $\mathrm{opt}/(\varepsilon\Phi)$, where $\Phi$ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost ($\mathrm{opt}$). 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 $\varepsilon$, and the number of steps depends logarithmically on $1/\varepsilon$ and quadratically on $\mathrm{opt}/\Phi$

Algorithmic Results for Clustering and Refined Physarum Analysis

In the first part of this thesis, we study the Binary $\ell_0$-Rank-$k$ problem which given a binary matrix $A$ and a positive integer $k$, seeks to find a rank-$k$ binary matrix $B$ minimizing the number of non-zero entries of $A-B$. A central open question is whether this problem admits a polynomial time approximation scheme. We give an affirmative answer to this question by designing the first randomized almost-linear time approximation scheme for constant $k$ over the reals, $\mathbb{F}_2$, and the Boolean semiring. In addition, we give novel algorithms for important variants of $\ell_0$-low rank approximation. The second part of this dissertation, studies a popular and successful heuristic, known as Approximate Spectral Clustering (ASC), for partitioning the nodes of a graph $G$ into clusters with small conductance. We give a comprehensive analysis, showing that ASC runs efficiently and yields a good approximation of an optimal $k$-way node partition of $G$. In the final part of this thesis, we present two results on slime mold computations: i) the continuous undirected Physarum dynamics converges for undirected linear programs with a non-negative cost vector; and ii) for the discrete directed Physarum dynamics, we give a refined analysis that yields strengthened and close to optimal convergence rate bounds, and shows that the model can be initialized with any strongly dominating point.Im ersten Teil dieser Arbeit untersuchen wir das Binary $\ell_0$-Rank-$k$ Problem. Hier sind eine bin{\"a}re Matrix $A$ und eine positive ganze Zahl $k$ gegeben und gesucht wird eine bin{\"a}re Matrix $B$ mit Rang $k$, welche die Anzahl von nicht null Eintr{\"a}gen in $A-B$ minimiert. Wir stellen das erste randomisierte, nahezu lineare Aproximationsschema vor konstantes $k$ {\"u}ber die reellen Zahlen, $\mathbb{F}_2$ und den Booleschen Semiring. Zus{\"a}tzlich erzielen wir neue Algorithmen f{\"u}r wichtige Varianten der $\ell_0$-low rank Approximation. Der zweite Teil dieser Dissertation besch{\"a}ftigt sich mit einer beliebten und erfolgreichen Heuristik, die unter dem Namen Approximate Spectral Cluster (ASC) bekannt ist. ASC partitioniert die Knoten eines gegeben Graphen $G$ in Cluster kleiner Conductance. Wir geben eine umfassende Analyse von ASC, die zeigt, dass ASC eine effiziente Laufzeit besitzt und eine gute Approximation einer optimale $k$-Weg-Knoten Partition f{\"u}r $G$ berechnet. Im letzten Teil dieser Dissertation pr{\"a}sentieren wir zwei Ergebnisse {\"u}ber Berechnungen mit Hilfe von Schleimpilzen: i) die kontinuierliche ungerichtete Physarum Dynamik konvergiert f{\"u}r ungerichtete lineare Programme mit einem nicht negativen Kostenvektor; und ii) f{\"u}r die diskrete gerichtete Physikum Dynamik geben wir eine verfeinerte Analyse, die st{\"a}rkere und beinahe optimale Schranken f{\"u}r ihre Konvergenzraten liefert und zeigt, dass das Model mit einem beliebigen stark dominierender Punkt initialisiert werden kann

Physarum Multi-Commodity Flow Dynamics

In wet-lab experiments \cite{Nakagaki-Yamada-Toth,Tero-Takagi-etal}, the slime mold Physarum polycephalum has demonstrated its ability to solve shortest path problems and to design efficient networks, see Figure \ref{Wet-Lab Experiments} for illustrations. Physarum polycephalum is a slime mold in the Mycetozoa group. For the shortest path problem, a mathematical model for the evolution of the slime was proposed in \cite{Tero-Kobayashi-Nakagaki} and its biological relevance was argued. The model was shown to solve shortest path problems, first in computer simulations and then by mathematical proof. It was later shown that the slime mold dynamics can solve more general linear programs and that many variants of the dynamics have similar convergence behavior. In this paper, we introduce a dynamics for the network design problem. We formulate network design as the problem of constructing a network that efficiently supports a multi-commodity flow problem. We investigate the dynamics in computer simulations and analytically. The simulations show that the dynamics is able to construct efficient and elegant networks. In the theoretical part we show that the dynamics minimizes an objective combining the cost of the network and the cost of routing the demands through the network. We also give alternative characterization of the optimum solution

A Laplacian Approach to ℓ1\ell_1-Norm Minimization

We propose a novel differentiable reformulation of the linearly-constrained $\ell_1$ minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based methods for the solution of $\ell_1$ minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The results can be seen as bounds on the complexity of iteratively reweighted least squares (IRLS) type methods of basis pursuit

Two results on slime mold computations

We present two results on slime mold computations. In wet-lab experiments by Nakagaki et al. (2000) [1] 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 (Tero et al., 2007) [3]. It was shown that the process convergences to the shortest path (Bonifaci et al., 2012) [5] 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 (Straszak and Vishnoi, 2016) [14]. 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