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 $\mathrm{opt}/\Phi$

On the Convergence Time of a Natural Dynamics for Linear Programming

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 LP instances. 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

Dynamic Resource Allocation for Efficient Sharing of Services from Heterogeneous Autonomous Vehicles

A novel dynamic resource allocation model is introduced for efficient sharing of services provided by ad hoc assemblies of heterogeneous autonomous vehicles. A key contribution is the provision of capability to dynamically select sensors and platforms within constraints imposed by time dependencies, refueling, and transportation services. The problem is modeled as a connected network of nodes and formulated as an integer linear program. Solution fitness is prioritized over computation time. Simulation results of an illustrative scenario are used to demonstrate the ability of the model to plan for sensor selection, refueling, collaboration, and cooperation between heterogeneous resources. Prioritization of operational cost leads to missions that use cheaper resources but take longer to complete. Prioritization of completion time leads to shorter missions at the expense of increased overall resource cost. Missions can be successfully replanned through dynamic reallocation of new requests during a mission. Monte Carlo studies on systems of increasing complexity show that good solutions can be obtained using low time resolutions, with small time windows at a relatively low computational cost. In comparison with other approaches, the developed integer linear program model provides best solutions at the expense of longer computation time

Rough-Cut Capacity Planning in Multimodal Freight Transportation Networks

A main challenge in transporting cargo for United States Transportation Command (USTRANSCOM) is in mode selection or integration. Demand for cargo is time sensitive and must be fulfilled by an established due date. Since these due dates are often inflexible, commercial carriers are used at an enormous expense, in order to fill the gap in organic transportation asset capacity. This dissertation develops a new methodology for transportation capacity assignment to routes based on the Resource Constrained Shortest Path Problem (RCSP). Routes can be single or multimodal depending on the characteristics of the network, delivery timeline, modal capacities, and costs. The difficulty of the RCSP requires use of metaheuristics to produce solutions. An Ant Colony System to solve the RCSP is developed in this dissertation. Finally, a method for generating near Pareto optimal solutions with respect to the objectives of cost and time is developed

Random Neural Networks and Optimisation

In this thesis we introduce new models and learning algorithms for the Random Neural Network (RNN), and we develop RNN-based and other approaches for the solution of emergency management optimisation problems. With respect to RNN developments, two novel supervised learning algorithms are proposed. The first, is a gradient descent algorithm for an RNN extension model that we have introduced, the RNN with synchronised interactions (RNNSI), which was inspired from the synchronised firing activity observed in brain neural circuits. The second algorithm is based on modelling the signal-flow equations in RNN as a nonnegative least squares (NNLS) problem. NNLS is solved using a limited-memory quasi-Newton algorithm specifically designed for the RNN case. Regarding the investigation of emergency management optimisation problems, we examine combinatorial assignment problems that require fast, distributed and close to optimal solution, under information uncertainty. We consider three different problems with the above characteristics associated with the assignment of emergency units to incidents with injured civilians (AEUI), the assignment of assets to tasks under execution uncertainty (ATAU), and the deployment of a robotic network to establish communication with trapped civilians (DRNCTC). AEUI is solved by training an RNN tool with instances of the optimisation problem and then using the trained RNN for decision making; training is achieved using the developed learning algorithms. For the solution of ATAU problem, we introduce two different approaches. The first is based on mapping parameters of the optimisation problem to RNN parameters, and the second on solving a sequence of minimum cost flow problems on appropriately constructed networks with estimated arc costs. For the exact solution of DRNCTC problem, we develop a mixed-integer linear programming formulation, which is based on network flows. Finally, we design and implement distributed heuristic algorithms for the deployment of robots when the civilian locations are known or uncertain

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

Coral Reef Islands and Their Problems

Small islands and in particular those more or less dependent on their reefs as a resource were often self-contained units, maintaining a fragile equilibrium in which even small changes could and have wrought fatal disturbances. The pressures of the modern world have endangered these vulnerable units economically, socially, and biologically. In all instances the \u27population factor has played an important role. Coral reef islands appear to be more susceptible to the side effects of rapidly growing modernization than other types of islands. No clear answer or solution to this problem can be given but recognition of the threats to them may bring the remedies closer