Choosing the Right Algorithm With Hints From Complexity Theory

Choosing a suitable algorithm from the myriads of different search heuristics is difficult when faced with a novel optimization problem. In this work, we argue that the purely academic question of what could be the best possible algorithm in a certain broad class of black-box optimizers can give fruitful indications in which direction to search for good established optimization heuristics. We demonstrate this approach on the recently proposed DLB benchmark, for which the only known results are O(n^3) runtimes for several classic evolutionary algorithms and an O(n^2 log n) runtime for an estimation-of-distribution algorithm. Our finding that the unary unbiased black-box complexity is only O(n^2) suggests the Metropolis algorithm as an interesting candidate and we prove that it solves the DLB problem in quadratic time. Since we also prove that better runtimes cannot be obtained in the class of unary unbiased algorithms, we shift our attention to algorithms that use the information of more parents to generate new solutions. An artificial algorithm of this type having an O(n log n) runtime leads to the result that the significance-based compact genetic algorithm (sig-cGA) can solve the DLB problem also in time O(n log n). Our experiments show a remarkably good performance of the Metropolis algorithm, clearly the best of all algorithms regarded for reasonable problem sizes.

On equilibrium Metropolis simulations on self-organized urban street networks

Urban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating system that preserves on average some amount of information. Monte Carlo methods that can further this perspective are cruelly missing. Here we adapt to self-organized urban street networks the Metropolis algorithm. The “coming to equilibrium” distribution is established with MaxEnt by taking scale-freeness as prior hypothesis along with symmetry-conservation arguments. The equilibrium parameter is the scaling; its concomitant extensive quantity is, assuming our lack of knowledge, an amount of information. To design an ergodic dynamics, we disentangle the state-of-the-art street generating paradigms based on non-overlapping walks into layout-at-junction dynamics. Our adaptation reminisces the single-spin-flip Metropolis algorithm for Ising models. We thus expect Metropolis simulations to reveal that self-organized urban street networks, besides sustaining scale-freeness over a wide range of scalings, undergo a crossover as scaling varies—literature argues for a small-world crossover. Simulations for Central London are consistent against the state-of-the-art outputs over a realistic range of scaling exponents. Our illustrative Watts–Strogatz phase diagram with scaling as rewiring parameter demonstrates a small-world crossover curving within the realistic window 2–3; it also shows that the state-of-the-art outputs underlie relatively large worlds. Our Metropolis adaptation to self-organized urban street networks thusly appears as a scaling variant of the Watts–Strogatz model. Such insights may ultimately allow the urban profession to anticipate self-organization or unplanned evolution of urban street networks.

Effect of surface charge self-organization on gate-induced 2D electron and hole systems

A simple model has been suggested for describing self-organization of localized charges and quantum scattering in undoped GaAs/AlGaAs structures with 2D electron or hole gas created by applying respective gate bias. It has been assumed that these metal / dielectric / undoped semiconductor structures exhibit predominant carrier scattering at localized surface charges which can be located at any point of the plane imitating the GaAs / dielectric interface. The suggested model considers all these surface charges and respective image charges in metallic gate as a closed thermostated system. Electrostatic self-organization in this system has been studied numerically for thermodynamic equilibrium states using the Metropolis algorithm over a wide temperature range. We show that at T > 100 K a simple formula derived from the theory of single-component 2D plasma yields virtually the same behavior of structural factor at small wave numbers as the one given by the Metropolis algorithm. The scattering times of gate-induced carriers are described with formulas in which the structural factor characterizes frozen disorder in the system. The main contribution in these formulas is due to behavior of the structural factor at small wave numbers. Calculation using these formulas for the case of disorder corresponding to infinite T has yielded 2–3 times lower scattering times than experimentally obtained ones. We have found that the theory agrees with experiment at disorder freezing temperatures T ≈ 1000 K for 2D electron gas specimen and T ≈ 700 K for 2D hole gas specimen. These figures are the upper estimates of freezing temperature for test structures since the model ignores all the disorder factors except temperature.

Asteroid mass estimation with the robust adaptive Metropolis algorithm

Context. The bulk density of an asteroid informs us about its interior structure and composition. To constrain the bulk density, one needs an estimated mass of the asteroid. The mass is estimated by analyzing an asteroid’s gravitational interaction with another object, such as another asteroid during a close encounter. An estimate for the mass has typically been obtained with linearized least-squares methods, despite the fact that this family of methods is not able to properly describe non-Gaussian parameter distributions. In addition, the uncertainties reported for asteroid masses in the literature are sometimes inconsistent with each other and are suspected to be unrealistically low. Aims. We aim to present a Markov-chain Monte Carlo (MCMC) algorithm for the asteroid mass estimation problem based on asteroid-asteroid close encounters. We verify that our algorithm works correctly by applying it to synthetic data sets. We use astrometry available through the Minor Planet Center to estimate masses for a select few example cases and compare our results with results reported in the literature. Methods. Our mass-estimation method is based on the robust adaptive Metropolis algorithm that has been implemented into the OpenOrb asteroid orbit computation software. Our method has the built-in capability to analyze multiple perturbing asteroids and test asteroids simultaneously. Results. We find that our mass estimates for the synthetic data sets are fully consistent with the ground truth. The nominal masses for real example cases typically agree with the literature but tend to have greater uncertainties than what is reported in recent literature. Possible reasons for this include different astrometric data sets and weights, different test asteroids, different force models or different algorithms. For (16) Psyche, the target of NASA’s Psyche mission, our maximum likelihood mass is approximately 55% of what is reported in the literature. Such a low mass would imply that the bulk density is significantly lower than previously expected and thus disagrees with the theory of (16) Psyche being the metallic core of a protoplanet. We do, however, note that masses reported in recent literature remain within our 3-sigma limits. Results. The new MCMC mass-estimation algorithm performs as expected, but a rigorous comparison with results from a least-squares algorithm with the exact same data set remains to be done. The matters of uncertainties in comparison with other algorithms and correlations of observations also warrant further investigation.