Asymptotic Ratio Research Articles

Approximation Algorithms for Multitasking Scheduling Problems

In this work, we incorporate human factors and real-life operations into newly proposed multitasking scheduling problems with periodic shift activities. It is motivated by personnel resource scheduling with periodic work shifts under the requirement of providing continuous service to customers. We model the problem as two identical parallel machine scheduling with complementary non-available time periods, and consider two models with the objectives of the makespan, i.e. the maximum completion time and respectively the total completion time. We then prove that the Greedy algorithm and SPT rule are of asymptotic and parametric approximation ratios for the two models, respectively.

Approximation Algorithms for Training One-Node ReLU Neural Networks

Training a one-node neural network with the ReLU activation function via optimization, which we refer to as the ON-ReLU problem, is a fundamental problem in machine learning. In this paper, we begin by proving the NP-hardness of the ON-ReLU problem. We then present an approximation algorithm to solve the ON-ReLU problem, whose running time is $\mathcal {O}(n^k)$ where $n$ is the number of samples, and $k$ is a predefined integral constant as an algorithm parameter. We analyze the performance of this algorithm under two regimes and show that: (1) given any arbitrary set of training samples, the algorithm guarantees an $(n/k)$ -approximation for the ON-ReLU problem – to the best of our knowledge, this is the first time that an algorithm guarantees an approximation ratio for arbitrary data scenario; thus, in the ideal case (i.e., when the training error is zero) the approximation algorithm achieves the globally optimal solution for the ON-ReLU problem; and (2) given training sample with Gaussian noise, the same approximation algorithm achieves a much better asymptotic approximation ratio which is independent of the number of samples $n$ . Extensive numerical studies show that our approximation algorithm can perform better than the gradient descent algorithm. Our numerical results also show that the solution of the approximation algorithm can provide a good initialization for gradient descent, which can significantly improve the performance.

On Online Algorithms for Bin, Strip, and Box Packing, and Their Worst-Case and Average-Case Analysis

In this survey, we consider online algorithms for bin packing and strip packing problems, as well as their generalizations (multidimensional bin packing, multiple strip packing, and packing into strips of different widths). For the latter problem, only the worst-case analysis is described; for the other problems, both the worst-case and average-case (probabilistic) asymptotic ratios are presented. The best lower and upper bounds are considered. Basic algorithms and methods for their analysis are discussed.

Low-Frequency Excess Noise Ratio Approximation for Avalanche Noise Diodes

This letter deals with microwave noise diodes in avalanche regime. Starting from the model proposed by Hines and Gliden in 1966, the asymptotic excess noise ratio (ENR) expression is derived for $\omega \rightarrow 0$ . The developed theory is then validated against ENR measurements of three noise diodes already published in the literature. The agreement between theory and measurements is good and the obtained formula also predicts the $1/I_{0}$ behavior of the ENR at low frequencies. This study is relevant because it simplifies the experimental determination of the average time between two ionizations (the Hines $\tau _{x}$ model parameter) by means of low-frequency noise measurements only.

A class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree

In this paper, we first introduce the asymptotic logarithmic likelihood ratio as a measure of the deviation between the arbitrary random fields and the bifurcating Markov chain on a binary tree. Then a class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree is established by constructing a nonnegative martingale. As corollaries, we obtain the strong law of large numbers (SLLN) and the asymptotic equipartition property (AEP) for the bifurcating Markov chains indexed by a binary tree.

Double Asymptotic for Random Walks on Hypercubes

We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the asymptotic ratio of the two parameters, they converge towards either a Brownian motion, an Ornstein-Uhlenbeck process or an i.i.d. collection of Gaussian variables.

A new lower bound on the price of anarchy of selfish bin packing

In standard bin packing, the goal is to partition or pack items with positive sizes of at most 1 into a minimum number of subsets, called bins, each of a total size no larger than 1. We study bin packing games. In these games, given a valid partition of the items, each item has a cost associated with it, based on the partition and on its size. Specifically, the cost of an item is the ratio between its size and the total size of items packed into its bin, that is, the cost sharing is in proportion to item sizes. We study pure Nash equilibria, which exist for all such games, and prove a new lower bound on the price of anarchy, which is the asymptotic worst-case ratio between the cost of the worst Nash equilibrium and a socially optimal packing.

QoS-Constrained Semi-Persistent Scheduling of Machine-Type Communications in Cellular Networks

The dramatic growth of machine-to-machine (M2M) communication in cellular networks brings the challenge of satisfying the quality of service (QoS) requirements of a large number of M2M devices with limited radio resources. In this paper, we propose an optimization framework for the semi-persistent scheduling of M2M transmissions based on the exploitation of their periodicity with the goal of reducing the overhead of the signaling required for connection initiation and scheduling. The goal of the optimization problem is to minimize the number of frequency bands used by the M2M devices to allow fair resource allocation of newly joining M2M and human-to-human communications. The constraints of the problem are delay and periodicity requirements of the M2M devices. We first prove that the optimization problem is NP-hard and then propose a polynomial-time heuristic algorithm employing a fixed priority assignment according to the QoS characteristics of the devices. We show that this heuristic algorithm provides an asymptotic approximation ratio of 2.33 to the optimal solution for the case where the delay tolerances of the devices are equal to their periods. Through extensive simulations, we demonstrate that the proposed algorithm performs better than the existing algorithms in terms of frequency band usage and schedulability.

Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes

We develop CIATA, a combined inversion-and-thinning approach for modeling a nonstationary non-Poisson process (NNPP), where the target arrival process is described by a given rate function and its associated mean-value function together with a given asymptotic variance-to-mean (dispersion) ratio. CIATA is based on the following: (i) a piecewise-constant majorizing rate function that closely approximates the given rate function from above; (ii) the associated piecewise-linear majorizing mean-value function; and (iii) an equilibrium renewal process (ERP) whose noninitial interrenewal times have mean 1 and variance equal to the given dispersion ratio. Transforming the ERP by the inverse of the majorizing mean-value function yields a majorizing NNPP whose arrival epochs are then thinned to deliver an NNPP having the specified properties. CIATA-Ph is a simulation algorithm that implements this approach based on an ERP whose noninitial interrenewal times have a phase-type distribution. Supporting theorems establish that CIATA-Ph can generate an NNPP having the desired mean-value function and asymptotic dispersion ratio. Extensive simulation experiments substantiated the effectiveness of CIATA-Ph with various rate functions and dispersion ratios. In all cases, we found approximate convergence of the dispersion ratio to its asymptotic value beyond a relatively short warm-up period. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0828 .

Asymptotic analysis of tail distortion risk measure under the framework of multivariate regular variation

Under the framework of multivariate regular variation, we obtain the asymptotic ratio between the tail distortion risk measure for portfolio loss and the sum of value-at-risk for single loss by a different method from the one in Zhu and Li when the confidence level tends to one. In order to illustrate the derived result, a relevant example is given and the corresponding numerical simulation is also carried out.

Correlation diagnosis method for heterogeneous Monte Carlo eigenvalue simulations based on a diffusion approximation

This paper presents a predictive model of the correlation level present in tally regions of any mesh size without requiring extensive high accuracy discrete tallies. This model provides a quick indication of the accuracy of the variance estimated from uncorrelated statistics. The prediction is developed from the generalization of the Multitype Branching Processes (MBP) model to a continuous space and solving correlation coefficients (and thus variance correction factors) exactly in an equivalent homogeneous problem. The real heterogeneous problem is homogenized in a way that preserves neutron migration area. Correlation coefficients and the asymptotic variance underestimation ratio of the homogenized problem provide a rough estimate of the correlations observed in the real heterogeneous problem for a given tally region size.

Integral of Scalar Curvature on Non-parabolic Manifolds

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold \((M^3, g)\) with non-negative Ricci curvature, the asymptotic- weighted scaling invariant integral of scalar curvature has an explicit bound in the form of asymptotic volume ratio.

Integrated optimization of material supplying, manufacturing, and product distribution: Models and fast algorithms

Motivated by applications in electroplating industry, we study an integrated optimization problem of production and logistics for a manufacturer and a third-party logistics (3PL) provider. The objective is to minimize the total cost which includes costs in transportation and inventory of materials, manufacturing, and inventory and outbound distribution of products. The materials are semi-products supplied by a single supplier, and the transportation of materials are accomplished by a 3PL provider. Semi-products may have different sizes and processing times, and they are processed on a batch-processing facility by the manufacturer. Outbound distribution of final products are accomplished by the same 3PL provider. We consider three different models and propose fast algorithms to solve each model. In the first model, semi-products have identical sizes and an optimal algorithm is proposed. In the second model, semi-products have identical processing times and an approximation algorithm is proposed. The algorithm has absolute and asymptotic worst case ratios of 1.5 and 1.223, respectively. In the third model, semi-products have arbitrary sizes and processing times and an approximation algorithm is proposed. The absolute and asymptotic worst case ratios of the algorithm are 2.181 and 2, respectively. The running time of the optimal algorithm of the first model is O(nlog n), while that of the approximation algorithms are O(n2).

Schedules for Self-monitoring Blood Pressure: A Systematic Review.

BACKGROUNDSelf-monitoring of blood pressure better predicts prognosis than clinic measurement, is popular with patients, and endorsed in hypertension guidelines. However, there is uncertainty over the optimal self-monitoring schedule. We therefore aimed to determine the optimum schedule to predict future cardiovascular events and determine “true” underlying blood pressure.METHODSSix electronic databases were searched from November 2009 (updating a National Institute for Health and Care Excellence [NICE] systematic review) to April 2017. Studies that compared aspects of self-monitoring schedules to either prognosis or reliability/reproducibility in hypertensive adults were included. Data on study and population characteristics, self-monitoring regime, and outcomes were extracted by 2 reviewers independently.RESULTSFrom 5,164 unique articles identified, 25 met the inclusion criteria. Twelve studies were included from the original NICE review, making a total of 37 studies. Increasing the number of days of measurement improved prognostic power: 72%–91% of the theoretical maximum predictive value (asymptotic maximum hazard ratio) was reached by 3 days and 86%–96% by 7 days. Increasing beyond 3 days of measurement did not result in better correlation with ambulatory monitoring. There was no convincing evidence that the timing or number of readings per day had an effect, or that ignoring the first day’s measurement was necessary.CONCLUSIONSHome blood pressure should be measured for 3 days, increased to 7 only when mean blood pressure is close to a diagnostic or treatment threshold. Other aspects of a monitoring schedule can be flexible to facilitate patient uptake of and adherence with self-monitoring.

Selection of quasi-stationary states in the Navier–Stokes equation on the torus

The two dimensional incompressible Navier–Stokes equation on with , periodic boundary conditions, and viscosity is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if , while the dipole will be selected if . Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of δ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all , there is an initial time period in which the high modes decay at the rapid rate , while the low modes evolve at the slower rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.

Cooling phonon modes of a Bose condensate with uniform few body losses

We present a general analysis of the cooling produced by losses on condensates or quasi-condensates. We study how the occupations of the collective phonon modes evolve in time, assuming that the loss process is slow enough so that each mode adiabatically follows the decrease of the mean density. The theory is valid for any loss process whose rate is proportional to the jjth power of the density, but otherwise spatially uniform. We cover both homogeneous gases and systems confined in a smooth potential. For a low-dimensional gas, we can take into account the modified equation of state due to the broadening of the cloud width along the tightly confined directions, which occurs for large interactions. We find that at large times, the temperature decreases proportionally to the energy scale mc^2mc2, where mm is the mass of the particles and cc the sound velocity. We compute the asymptotic ratio of these two quantities for different limiting cases: a homogeneous gas in any dimension and a one-dimensional gas in a harmonic trap.

Quality of strong equilibria for selfish bin packing with uniform cost sharing

The bin packing problem deals with packing items of sizes no larger than 1 into unit capacity bins. Here, we analyze a class of bin packing games where the cost of an item is 1 over the total number of items packed into its bin, which is a bin packing congestion game. We study strong equilibria and find the tight values of the SPoA and SPoS, that is, asymptotic approximation ratios of the worst and best strong equilibria. We show that these values are approximately 1.69103 and 1.611824, respectively. In particular, we observe that the two values are not equal, showing a difference from other known kinds of cost sharing approaches.

Asymptotic Likelihood Ratio Test for the Parameter of Inflation in ZIP

In many cases, a large proportion of count data are zeros and the zero-inflated models such as the zero-inflated Poisson may be appropriate. This zero-inflated phenomenon is a specific cause of over-dispersion, and zero-inflated Poisson model has been proposed for accommodating zero-inflated data. This study proposes an asymptotic likelihood ratio test to compare the zero-inflated Poisson model versus the standard Poisson model. The advantage of the given test is that without fitting the zero-inflated Poisson or the Poisson model to the data it can be determined which of the Poisson or the zero-inflated Poisson model is appropriate for the given data. The performance of the given test is evaluated, and its results show that the power of the test is remarkable especially for values in parameter space for which the other tests have low power. Some interesting applications of proposed test have been done, which show better result in comparison with the previous tests.

New Sets of Even-Length Binary Z-Complementary Pairs With Asymptotic ZCZ Ratio of <inline-formula> <tex-math notation="LaTeX">$3/4$</tex-math> </inline-formula>

This letter is focused on increasing the zero correlation zone (ZCZ) of even-length binary Z-complementary pairs (EB-ZCPs). Till date, the maximum ZCZ ratio (i.e., ZCZ width over the sequence length) for systematically constructed EB-ZCPs is $2/3$ . In this letter, we give a construction of EB-ZCPs with lengths $2^{\alpha +2} 10^\beta 26^\gamma +2$ (where $\alpha$ , $\beta$ , and $\gamma$ are nonnegative integers) and ZCZ widths $3 \times 2^\alpha 10^\beta 26^\gamma +1$ , thus achieving asymptotic ZCZ ratio of $3/4$ . The proposed EB-ZCPs are constructed via proper insertion of concatenated odd-length binary ZCPs. The ZCZ width is proved by exploiting several newly identified intrinsic structure properties of binary Golay complementary pairs, obtained from Turyn's method. The proposed EB-ZCPs have aperiodic autocorrelation sums (AACS) magnitude of 4 outside the ZCZ region (except for the last time-shift taking AACS value of zero).

Optimization of batch operations with a truncated batch-position-based learning effect

Motivated by applications in art tile manufacturing and metal working industries, we study the optimization problem with a truncated batch-position-based learning effect. In production, a set of semi-products need to be processed on a single batch facility which has a fixed capacity. Several semi-products can be processed together in one batch if their total size does not exceed the facility capacity. We consider a truncated batch-position-based learning effect which is a typical behavior of workers. During the learning period, the worker can finish the task more and more quickly because of learning effects. After the learning period, the worker reaches the best ability and the ability keeps steady. Then we consider two models of manufacturing with batch operations. In the first model, semi-products have identical sizes and we propose an optimal algorithm with time complexity of O(nlog n). In the second model, semi-products have arbitrary sizes which are proportional to their processing times and the model is shown to be NP-hard in the strong sense. We propose two types of learning effects including fast and slow truncated batch-position-based learning effects. Then we propose an approximation algorithm with an absolute and asymptotic worst-case ratio less than 2. Finally, we conduct computational experiments and the results show the effectiveness of our algorithms. We also provide managerial insights and detailed suggestions for decision makers of manufacturing companies based on our results.