Approximation Scheme Research Articles

Envy-Free Division of Multilayered Cakes

Dividing a multilayered cake under nonoverlapping constraints captures several scenarios (e.g., allocating multiple facilities over time where each agent can utilize at most one facility simultaneously). We establish the existence of an envy-free multidivision that is nonoverlapping and contiguous within each layer when the number of agents is a prime power, solving partially an open question by Hosseini et al. [Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Proc. 29th Internat. Joint Conf. Artificial Intelligence (IJCAI), 182–188; Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Preprint, submitted April 28, http://arxiv.org/abs/2004.13397 ]. Our approach follows an idea proposed by Jojić et al. [Jojić D, Panina G, Živaljević R (2021) Splitting necklaces, with constraints. SIAM J. Discrete Math. 35(2):1268–1286] for envy-free divisions, relying on a general fixed-point theorem. We further design a fully polynomial-time approximation scheme for the two-layer, three-agent case, with monotone preferences. All results are actually established for divisions among groups of almost the same size. In the one-layer, three-group case, our algorithm is able to deal with any predetermined sizes, still with monotone preferences. For three groups, this provides an algorithmic version of a recent theorem by Segal-Halevi and Suksompong [Segal-Halevi E, Suksompong W (2021) How to cut a cake fairly: A generalization to groups. Amer. Math. Monthly 128(1):79–83]. Funding: This work was partially supported by the Japan Science and Technology Agency [Grant JPMJPR20C], Fusion Oriented REsearch for disruptive Science and Technology [Grant JPMJFR226O], and Exploratory Research for Advanced Technology [Grant JPMJER2301].

A truthful randomized mechanism for task allocation with multi-attributes in mobile edge computing

Mobile Edge Computing (MEC) aims at decreasing the response time and energy consumption of running mobile applications by offloading the tasks of mobile devices (MDs) to the MEC servers located at the edge of the network. The demands are multi-attribute, where the distances between MDs and access points lead to differences in required resources and transmission energy consumption. Unfortunately, the existing works have not considered both task allocation and energy consumption problems. Motivated by this, this paper considers the problem of task allocation with multi-attributes, where the problem consists of the winner determination and offloading decision problems. First, the problem is formulated as the auction-based model to provide flexible service. Then, a randomized mechanism is designed and is truthful in expectation. This drives the system into an equilibrium where no MD has incentives to increase the utility by declaring an untrue value. In addition, an approximation algorithm is proposed to minimize remote energy consumption and is a polynomial-time approximation scheme. Therefore, it achieves a tradeoff between optimality loss and time complexity. Simulation results reveal that the proposed mechanism gets the near-optimal allocation. Furthermore, compared with the baseline methods, the proposed mechanism can effectively increase social welfare and bring higher revenue to edge server providers.

Locally verifiable approximate multi-member quantum threshold aggregation digital signature scheme

Locally verifiable aggregate signature primitives can reduce the complexity of aggregate signature verification by computing locally open algorithms to generate auxiliary parameters. However, the breakthrough results of quantum computers at this stage indicate that it will be possible for quantum computers to break through the security of traditional hardness-based aggregated signature schemes. In order to solve the above problems, this paper proposes for the first time a new locally verifiable class of multi-member quantum threshold aggregated digital signature scheme based on the property that the verification of quantum coset states is a projection on the trans-subspace. Combined with the idea of auxiliary parameter generation in traditional locally verifiable aggregated signatures, it makes the current stage of threshold quantum digital signatures realize the aggregated features, and reduces the complexity of the verification of aggregated signatures while realizing post-quantum security. In addition, the verification of the signature key (quantum state) of the signature members does not require measurement operations, and the generated signatures are classical, so the communication between the trusted third center (TC), the set of signature members, the classical digital signature verifier (CV), and the third-party trusted aggregation generator (TA) are all classical, simplifying the communication model. In the performance analysis we make this quantum aggregation signature scheme more flexible as well as less quantum state preparation compared to other schemes.

Local spline refinement driven by fault jump estimates for scattered data approximation

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

The Continuous-Time Joint Replenishment Problem: ϵ-Optimal Policies via Pairwise Alignment

The main contribution of this paper resides in developing a new algorithmic approach for addressing the continuous-time joint replenishment problem, termed [Formula: see text]-pairwise alignment. The latter mechanism, through which we synchronize multiple economic order quantity models, allows us to devise a purely combinatorial algorithm for efficiently approximating optimal policies within any degree of accuracy. As a result, our work constitutes the first quantitative improvement over power-of-2 policies, which have been state-of-the-art in this context since the mid-1980s. Moreover, in light of recent intractability results, by proposing an efficient polynomial-time approximation scheme for the joint replenishment problem, we resolve the long-standing open question regarding the computational complexity of this classical setting. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Israel Science Foundation [Grant 1407/20].

Smoothing and approximation of grassland fire loading data for engineering structures by Capped Extended Support Vector Regression

This research proposes a machine learning-aided data smoothing and approximation scheme to investigate grassland fire loading for engineering structures. The investigations on grassland fire loading have significant impacts on wildfire-resistant design, building codes, regulations, and the like in specific regions. Real-world data collection systems inevitably introduce imperfections, such as noise, outliers, and the like. Furthermore, the collected data are normally discretized on time and location, and thus the information between them is completely absent. Achieving a smooth fit to data and reducing these imperfections are desired to facilitate subsequent analyses. Accordingly, a supervised machine learning algorithm is developed to smooth and approximate grassland fire loading data. Within the proposed algorithm, a convex optimization programming is established, which provides theoretical support to the accuracy of the algorithm. Then, through the proposed scheme, the grassland fire loading data can be approximated into an explicit regression model, which continuously completes the missing information during the observations. Moreover, attributed to the sparsity feature of the developed algorithm, the model can be simplified in its expression to further improve its applicability in engineering. Finally, real-world experimental data and numerical simulation results are separately investigated to demonstrate the effectiveness of the proposed scheme.

Hybrid differential evolution algorithm for Falkner-Skan flow with rotation

A hybrid differential evolution algorithm is used in this work to study the rotating transport of Falkner-Skan flow. The problem is modeled as an equivalent optimization problem by using Padé rational approximation functions. The primary model of the governing partial differential equations is imposed to subjugate the error between profiles. A hybrid evolutionary algorithm based on differential evolution and a convergent version of Nelder-Mead direct search algorithm is employed to perform global exploratory search along with an enhanced exploitation to improve the accuracy of the proposed solution scheme. The resulting scheme is named as evolutionary Padé approximation (EPA) scheme. The performance of the proposed EPA scheme on the Falkner-Skan boundary value problem is investigated by considering various values of the rotation parameters. The developed optimizer in EPA scheme was able to minimize the residuals up to10−10. Results are displayed graphically in order to study the effect of various types of parameters. EPA scheme determined that angular velocity increases or decreases accordingly as fluid parameter (β) and the rotation parameter (λ) but shows inverse behavior with respect to power law index(n). Similarly, the response of velocity profile along y−axis was decreasing function of β as well as n but increasing function of λ. The performance of the proposed EPA scheme has been demonstrated by comparing results with a hybrid neural network scheme and found in excellent agreement.

Parameterised approximation of the fixation probability of the dominant mutation in the multi-type Moran process

The multi-type Moran process is an evolutionary process on a connected graph G in which each vertex has one of k types and, in each step, a vertex v is chosen to reproduce its type to one of its neighbours. The probability of a vertex v being chosen for reproduction is proportional to the fitness of the type of v. So far, the literature was almost solely concerned with the 2-type Moran process in which each vertex is either healthy (type 0) or a mutant (type 1), and the main problem of interest has been the (approximate) computation of the so-called fixation probability, i.e., the probability that eventually all vertices are mutants.In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected absorption time, i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.

Emergence of Koba-Nielsen-Olsen scaling in multiplicity distributions in jets produced at the LHC

In this work we study the multiplicity distributions (MDs) of charged particles within jets in proton-proton collisions, which were measured by the ATLAS Collaboration in 2011, 2016 and 2019. The first dataset refers to jets with smaller transverse momenta (4<pT<40 GeV) whereas the other two refer to higher pT jets (0.1<pT<2.5 TeV). We find that the lower pT set shows no sign of KNO scaling and that the higher pT sets gradually approach the scaling limit. For the lower pT set the mean multiplicity as a function of pT can be well-described by expressions derived from QCD with different approximation schemes. For higher (>500 GeV) values of pT these expressions significantly overshoot the data. We show that the behavior of the MDs can be well-represented by a sub-Poisson distribution with energy dependent parameters. In the range 40<pT<100 GeV there is a transition from sub to super Poissonian behavior and the MD evolves to a geometric distribution, which shows KNO scaling. In this way we fit the MDs in all transverse momentum intervals with one single expression. We discuss the implications of this phenomenological finding. Published by the American Physical Society 2024

Estimation of equilibration time scales from nested fraction approximations.

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients or solving a system of coupled differential equationsin the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function and use it to estimate the equilibration time τ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of τ, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.

EXPRESS: Joint Assortment and Price Optimization with Multiple Purchases

In this paper, we incorporate the effects of multi-unit purchases into consumer choice models and investigate the associated joint assortment and pricing problems. Under the proposed two-stage choice framework, consumers first form a consideration set, and then select a product with the optimal quantity from the consideration set to maximize their utility. For the joint assortment and pricing problem with homogeneous consumers, we find that a multi-purchase-willingness-ordered assortment is optimal under certain technical conditions, and propose a polynomial-time algorithm to find the optimal assortment and prices for general cases. We further show that the optimal prices and assortment size are not monotone in multi-purchase preference and multi-purchase resistance. For the joint assortment and pricing problem with heterogeneous consumers, we first prove that this problem is NP-hard. We then consider a scenario with discrete prices and we develop a fully polynomial-time approximation scheme by considering a finite number of consumer segments and nested consideration sets. We conduct an empirical study on the JD.com dataset and show that the new choice framework can improve model fitting and prediction accuracy compared with existing choice models with multiple purchases. We reveal that our model is especially suitable for datasets exhibiting a strong diminishing marginal effect for product variety. We further extend our analysis to price competition and establish the existence and uniqueness of a Nash equilibrium. For the joint optimization with fixed costs, we develop an algorithm that can find a solution guaranteeing at least half the optimal expected profit.

Novel efficient iterative schemes for linear finite element approximations of elliptic optimal control problems with integral constraint on the state

Centering on a distributed optimal control problem governed by elliptic equations with a unilateral integral constraint on the state, we recall the first-order optimality conditions to explore variational formulations. The classical linear finite element method is employed to approximate the equivalent formulae. Taking into account the distinctive structural characteristics of discretized schemes, we divide the discretized optimal conditions into two equivalent matrix schemes. Subsequently, we introduce an efficient iterative algorithm designed to solve the approximation schemes. We demonstrate the convergence of our iterative algorithm and investigate its robustness and uniform optimality under a specified constraint, particularly focusing on small regularization parameters. Finally, numerical tests are conducted using various h and α to illustrate the efficiency of our proposed algorithm. These tests also validate a tremendous reduction in the number of required iterations.

Efficient method for symmetric nonnegative matrix factorization with an approximate augmented Lagrangian scheme

In this paper, we propose an efficient method for solving symmetric nonnegative matrix factorization following an approximate augmented Lagrangian scheme. The augmented Lagrangian subproblem was solved column by column under the block coordinate descent (BCD) framework. In particular, we extend the recursive formula for rank-k nonnegative least squares problems by Chu et al. (2021) to subproblems generated by BCD framework in our method. Thereafter we derive the closed-form solution for k=4. Experiments for clustering show that the proposed method is noticeably efficient and achieves competitive performance compared with state-of-the-art methods.

Single machine scheduling with the total weighted late work and rejection cost

AbstractWe study a single machine scheduling problem with the total weighted late work and the total rejection cost. The late work of a job is the part of this job executed after its due date, and the rejection cost of a job is the fee of rejecting to process it. We consider a Pareto scheduling and two restricted scheduling problems. The Pareto scheduling problem aims to find all non‐dominated values of the total weighted late work and the total rejection cost. The restricted scheduling problems are dedicated to minimizing the total weighted late work with the total rejection cost not greater than a given threshold, and minimizing the total rejection cost with the total weighted late work not greater than a given threshold, respectively. For the Pareto scheduling problem, we prove that it is binary NP‐hard by providing a pseudo‐polynomial time algorithm, and give a fully polynomial time approximation scheme (FPTAS). For the restricted scheduling problems, we prove that there are no FPTASes unless , which answers an open problem. Moreover, we develop relaxed FPTASes for these two restricted scheduling problems.

Bounded mixed batch scheduling with job release dates and rejection

This paper investigates a bounded mixed batch scheduling problem with job release dates and rejection. The machine processes a batch containing several jobs that their number does not exceed the machine capacity. For the case with a certain release date, we present a polynomial-time exact algorithm. For the case with a constant number of release dates, a pseudo-polynomial-time exact algorithm is proposed. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.

Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach

Lévy processes time-changed by inverse subordinators have been intensively studied in the last years. Their importance in connection with non-local operators and semi-Markov dynamics is well understood, but, in our view, several questions remain open concerning the probabilistic structure of such processes. The time-changed Lévy processes are particularly useful to describe complex systems with fractional and/or anomalous dynamics. The purpose of our work is to analyze the features of the sample paths of such processes, focusing on a martingale-based approach. We introduce the fractional Poisson random measure as the main tool for dealing with the jump component of time-changed càdlàg processes. Further, the fractional random measure is an interesting and novel topic in itself, and thus, it is thoroughly analyzed in the paper. A central role in our analysis is then played by fractional Poisson integrals (involving the aforementioned fractional Poisson measure) which allow a useful description of the random jumps. We investigate these stochastic integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semimartingale representation of time-changed processes analogous to the celebrated Lévy–Itô decomposition. Finally, an approximation scheme of such random processes will be discussed.

A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

Advanced neural network approaches for coupled equations with fractional derivatives

We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods.Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.

On a hemivariational inequality with non-monotone operator

We investigate the existence of generalized variational solutions to abstract hemivariational inequality driven by a coercive and locally Lipschitz operator with no further monotonicity conditions imposed on it. A finite dimensional Ritz type approximation scheme is derived. Applications to competitive systems driven by the generalized ( p , q ) - Laplacian with convection are given.

Minimising the makespan on parallel identical machines with log-linear position-dependent processing times

This article studies the minimum makespan scheduling problem on parallel identical machines with the log-linear learning/ageing effect, also known as polynomial positional learning/ageing effect. To develop a Fully Polynomial Time Approximation Scheme to the problem, we start with an intermediate artificial variant that rounds the values to integers and restricts the solutions to instances sorted in the shortest/longest processing time order. To this end, we propose a dynamic programming algorithm and show that the difference between its returned value and the minimum makespan of the original problem is independent of the processing times. This then leads to an algorithm with provable guaranteed ϵ-additive approximation and pseudo-polynomial running time algorithm, resulting in the desired fully polynomial time approximation solution to the original, not restricted problem.