Approximation Algorithm Research Articles

Approximation algorithms for the airport and railway problem

Approximation algorithms for the airport and railway problem

Mutual and total mutual visibility in hypercube-like graphs

Let G be a graph and X⊆V(G). Then, vertices x and y of G are X-visible if there exists a shortest x,y-path where no internal vertices belong to X. The set X is a mutual-visibility set of G if every two vertices of X are X-visible, while X is a total mutual-visibility set if any two vertices from V(G) are X-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) μ(G) (resp. μt(G)) of G. It is known that computing μ(G) is an NP-complete problem, as well as μt(G). In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, Fibonacci cubes, cube-connected cycles, and butterflies). Concerning computing μ(G), we provide approximation algorithms for hypercubes, Fibonacci cubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing μt(G) (in the literature, already studied in hypercubes), whereas we obtain exact formulae for both cube-connected cycles and butterflies.

Model-free active noise control using second-order simultaneous perturbation stochastic approximation algorithm for periodic noise

Model-free active noise control (ANC) using simultaneous perturbation stochastic approximation (SPSA) algorithm has recently attracted interest of researchers since it avoids the problem of secondary path modeling (SPM). However, the existing SPSA-based ANC methods are all based on the first-order SPSA algorithm and the convergence rate of these methods are significantly lower than that of the Filtered-x least mean square (FxLMS) algorithm. In order to enhance the convergence performance, this paper introduces the second-order SPSA algorithm into the ANC system. During the implementation of the second-order SPSA algorithm, a novel method of matrix positive definite transformation based on eigenvalue decomposition (EVD) is proposed. Convergence performance and tracking performance of ANC using the second-order SPSA algorithm are compared with ANC using the FxLMS algorithm and the standard SPSA through simulation and experimental verification. Simulation and experimental verification results show that ANC using the second-order SPSA algorithm can not only achieve the convergence rate comparable to that of ANC using the FxLMS algorithm, but also track the secondary path change.

L-ASCRA: A Linearithmic Time Approximate Spectral Clustering Algorithm Using Topologically-Preserved Representatives

L-ASCRA: A Linearithmic Time Approximate Spectral Clustering Algorithm Using Topologically-Preserved Representatives

Approximation and Heuristic Algorithms for the Priority Facility Location Problem with Outliers

Approximation and Heuristic Algorithms for the Priority Facility Location Problem with Outliers

A novel design approach for structures with inerter systems based on non-stationary seismic excitations

Existing research predominantly focuses on developing design methods for the structural configuration of inerter systems. However, the authors find that these optimization methods often overlook the non-stationary characteristics associated with seismic response. Additionally, traditional approximation algorithms yield significant errors in non-stationary processes, resulting in inaccurate predictions of the structural responses with inerter systems, which compromises the effectiveness of the optimization designs. This paper proposes a novel approach that combines the parameters of the inerter system with a modified Vanmarcke approximation to obtain unified calculation formulas for inerter parameters. The accuracy and computational efficiency in solving spectral moments are crucial for the feasibility of this method. The study presents an efficient calculation method to accurately determine non-stationary response spectral moments of structures with inerter systems. Using the complex modal method (CMM) and pseudo-excitation method (PEM), a unified solution for structural responses under seismic actions is obtained. The quadratic decomposition of the power spectral density function (QD-PSDF) is employed to derive analytical solutions for response spectral moments and non-geometric spectral characteristics (NGSC). Substituting the obtained moment values into the parameter calculation formulas allows effective determination of the inerter system parameters. Finally, Finite Element Method (FEM) modeling verifies the validity and general applicability of this approach. The results demonstrate that the proposed method accounts for non-stationary characteristics of structures with inerter systems under seismic excitations, enhancing seismic design reliability and structural safety. Moreover, the obtained inerter system parameters at each floor effectively achieve the target performance while providing a broader range of design options for enhanced flexibility in system configuration.

A reliable strategy for establishment of an animal model of diabetic cardiomyopathy: Induction by a high-fat diet combined with single or multiple injections of low-dose streptozotocin

BackgroundDiabetic cardiomyopathy (DCM) is one of the leading causes of death in patients with diabetes mellitus (DM). This study aimed to identify a reliable method for establishing an animal model of DCM for investigation of new targets and treatments. MethodsEighty-four 4-week-old male Sprague–Dawley rats were randomly allocated to receive a normal diet or a high-fat diet (HFD) in an approximate ratio of 1:3. At 9 weeks of age, rats in the HFD group received streptozotocin (STZ) 30 mg/kg by intraperitoneal injection and rats in the control group received the same volume of buffer solution. The rodent model of DM was deemed to be successfully established when a random blood glucose measurement was >16.7 mmol/L on three consecutive occasions. If necessary, STZ was readministered. ResultsThree of the 64 rats in the HFD group died after a second STZ injection. DM was induced in 14, 39, and 8 rats after one, two, and three injections, respectively, with cumulative success rates of 21.9 %, 82.8 %, and 95.3 %. Three months later, the rats with DM showed persistent hyperglycemia and insulin resistance and developed histopathological changes indicating cardiac hypertrophy, myocardial fibrosis, and diastolic dysfunction. The metabolic and cardiac histopathological changes were consistent regardless of whether DM was induced by one, two, or three injections of STZ. ConclusionAn HFD combined with one or more intraperitoneal injections of low-dose STZ is a straightforward and reliable method for inducing DCM in rats. When a single dose of STZ fails to induce DM, repeated injections can be considered.

Optimizing Euclidean Distance Computation

This paper presents a comparative analysis of seventeen different approaches to optimizing Euclidean distance computations, which is a core mathematical operation that plays a critical role in a wide range of algorithms, particularly in machine learning and data analysis. The Euclidean distance, being a computational bottleneck in large-scale optimization problems, requires efficient computation techniques to improve the performance of various distance-dependent algorithms. To address this, several optimization strategies can be employed to accelerate distance computations. From spatial data structures and approximate nearest neighbor algorithms to dimensionality reduction, vectorization, and parallel computing, various approaches exist to accelerate Euclidean distance computation in different contexts. Such approaches are particularly important for speeding up key machine learning algorithms like K-means and K-nearest neighbors (KNNs). By understanding the trade-offs and assessing the effectiveness, complexity, and scalability of various optimization techniques, our findings help practitioners choose the most appropriate methods for improving Euclidean distance computations in specific contexts. These optimizations enable scalable and efficient processing for modern data-driven tasks, directly leading to reduced energy consumption and a minimized environmental impact.

Complexity issues concerning the quadruple Roman domination problem in graphs

Given a graph G with vertex set V(G), a mapping h:V(G)→{0,1,2,3,4,5} is called a quadruple Roman dominating function (4RDF) for G if it holds the following. Every vertex x such that h(x)∈{0,1,2,3} satisfies that h(N[x])=∑v∈N[x]h(v)≥|{y:y∈N(x)andh(y)≠0}|+4, where N(x) and N[x] stands for the open and closed neighborhood of x, respectively. The smallest possible weight ∑x∈V(G)h(x) among all possible 4RDFs h for G is the quadruple Roman domination number of G, denoted by γ[4R](G).This work is focused on complexity aspects for the problem of computing the value of this parameter for several graph classes. Specifically, it is shown that the decision problem concerning γ[4R](G) is NP-complete when restricted to star convex bipartite, comb convex bipartite, split and planar graphs. In contrast, it is also proved that such problem can be efficiently solved for threshold graphs where an exact solution is demonstrated, while for graphs having an efficient dominating set, tight upper and lower bounds in terms of the classical domination number are given. In addition, some approximation results to the problem are given. That is, we show that the problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless P=NP. An approximation algorithm for it is proposed, and its APX-completeness proved, whether graphs of maximum degree four are considered. Finally, an integer linear programming formulation for our problem is presented.

Towards proactive rumor control: When a budget constraint meets impression counts

The proliferation of rumors in online networks poses significant public safety risks and economic repercussions. Addressing this, we investigate the understudied aspect of rumor control: the interplay between influence block effect and user impression counts under budget constraints. We introduce two problem variants, RCIC and RCICB, tailored for diverse application contexts. Our study confronts two inherent challenges: the NP-hard nature of the problems and the non-submodularity of the influence block, which precludes direct greedy approaches. We develop a novel branch-and-bound framework for RCIC, achieving a (1−1/e−ϵ) approximation ratio, and enhance its efficacy with a progressive upper bound estimation, refining the ratio to (1−1/e−ϵ−ρ). Extending these techniques to RCICB, we attain approximation ratios of (12(1−1/e)−ϵ) and (12(1−1/e−ρ)−ϵ). We conduct experiments on real-world datasets to verify the efficiency, effectiveness, and scalability of our methods.

On the comparative utility of entropic learning versus deep learning for long-range ENSO prediction

Abstract This paper compares the ability of deep learning and entropic learning methods to predict the probability of the Niño3.4 index being above 0.4° (El Niño), below −0.4° (La Niña) or within both of these thresholds (neutral) at lead times of 3 up to 24 months. In particular, the performance, interpretability, and training cost of entropic learning methods, represented by the entropy-optimal Scalable Probabilistic Approximation (eSPA) algorithm, are compared with deep learning methods, represented by a Long Short-Term Memory (LSTM) classifier, trained on the same dataset. Using only monthly data derived from a surface-forced ocean model over the period 1958-2018, the problem manifests as a canonical small-data challenge. Relative to the LSTM model, eSPA exhibits equivalent or better performance in terms of area under the receiver operating characteristic curve (AUC) for all lead times with a smaller computational cost by 2-3 orders of magnitude. Comparisons of AUC with other state-of-the-art deep learning models in the literature show that eSPA appears to also be more accurate than these models across all three classes. Composite images are generated for each of the cluster centroids from each trained eSPA model at each lead time. At shorter lead times, the composite images for the most significant clusters correspond to patterns representing mature or emerging/declining El Niño or La Niña states, while at longer lead times they correspond to precursor states consisting of extra-tropical anomalies. Finally, modifications to the baseline dataset are explored, showing that improvements can be made in the parsimony of the trained eSPA model without sacrificing predictive power.

Two-Stage Estimation and Variance Modeling for Latency-Constrained Variational Quantum Algorithms

The quantum approximate optimization algorithm (QAOA) has enjoyed increasing attention in noisy, intermediate-scale quantum computing with its application to combinatorial optimization problems. QAOA has the potential to demonstrate a quantum advantage for NP-hard combinatorial optimization problems. As a hybrid quantum-classical algorithm, the classical component of QAOA resembles a simulation optimization problem in which the simulation outcomes are attainable only through a quantum computer. The simulation that derives from QAOA exhibits two unique features that can have a substantial impact on the optimization process: (i) the variance of the stochastic objective values typically decreases in proportion to the optimality gap, and (ii) querying samples from a quantum computer introduces an additional latency overhead. In this paper, we introduce a novel stochastic trust-region method derived from a derivative-free, adaptive sampling trust-region optimization method intended to efficiently solve the classical optimization problem in QAOA by explicitly taking into account the two mentioned characteristics. The key idea behind the proposed algorithm involves constructing two separate local models in each iteration: a model of the objective function and a model of the variance of the objective function. Exploiting the variance model allows us to restrict the number of communications with the quantum computer and also helps navigate the nonconvex objective landscapes typical in QAOA optimization problems. We numerically demonstrate the superiority of our proposed algorithm using the SimOpt library and Qiskit when we consider a metric of computational burden that explicitly accounts for communication costs. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers and the Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing program under contract number DE-AC02-06CH11357. Y. Ha and S. Shashaani also gratefully acknowledge the U.S. National Science Foundation Division of Civil, Mechanical and Manufacturing Innovation Grant CMMI-2226347 and the U.S. Office of Naval Research [Grant N000142412398]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0575 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0575 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Playing Divide-and-Choose Given Uncertain Preferences

We study the classic divide-and-choose method for equitably allocating divisible goods between two players who are rational, self-interested Bayesian agents. The players have additive values for the goods. The prior distributions of those values are common knowledge. We consider both the cases of independent values and values that are correlated across players (as occurs when there is a common-value component). We describe the structure of optimal divisions in the divide-and-choose game and identify several cases where it is possible to efficiently compute equilibria. An approximation algorithm is presented for the case when the distribution over the chooser’s value for each good follows a normal distribution, along with a randomized approximation algorithm for the case of uniform distributions over intervals. A mixture of analytic results and computational simulations illuminates several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, given unknown preferences, the divider has a compelling “diversification” incentive in creating the chooser’s two options. This incentive leads to multiple goods being divided at equilibrium, quite contrary to the divider’s optimal strategy when preferences are known. In many contexts, such as buy-and-sell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players’ levels of knowledge about each other’s values, the correlations between the players’ values, and the number of goods being divided. Under fairly mild assumptions, we show that the chooser is strictly better off for a small number of goods, whereas the divider is strictly better off for a large number of goods. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis. Funding: This work was supported by the Mossavar-Rahmani Center for Business and Government, Harvard University, and the National Science Foundation [Grant DGE1745303]. Any opinions, findings, and conclusions or recommendations expressed in this material reflect the views of the authors and not necessarily those of the National Science Foundation or the Mossavar-Rahmani Center. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.00350 .

LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism

We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA (LX-QAOA), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms.

Enhanced deterministic approximation algorithm for non-monotone submodular maximization under knapsack constraint with linear query complexity

Enhanced deterministic approximation algorithm for non-monotone submodular maximization under knapsack constraint with linear query complexity

Improved Guarantees for the A Priori TSP

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space (V, c) and an activation probability p(v) for each customer [Formula: see text]. We ask for a TSP tour T for V that minimizes the expected length after cutting T short by skipping the inactive customers. All known approximation algorithms select a nonempty subset S of the customers and construct a master route solution, consisting of a TSP tour for S and two edges connecting every customer [Formula: see text] to a nearest customer in S. We address the following questions. If we randomly sample the subset S, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling and less than 5.9 with a deterministic polynomial-time algorithm.

Fast and Globally Consistent Normal Orientation based on the Winding Number Normal Consistency

Estimating consistently oriented normals for point clouds enables a number of important applications in computer graphics such as surface reconstruction. While local normal estimation is possible with simple techniques like principal component analysis (PCA), orienting these normals to be globally consistent has been a notoriously difficult problem. Some recent methods exploit various properties of the winding number formula to achieve global consistency with state-of-the-art performance. Despite their exciting progress, these algorithms either have high space/time complexity, or do not produce accurate and consistently oriented normals for imperfect data. In this paper, we propose a novel property from the winding number formula, termed Winding Number Normal Consistency (WNNC ), to tackle this problem. The derived property is based on the simple observation that the normals (negative gradients) sampled from the winding number field should be codirectional to the normals used to compute the winding number field. Since the WNNC property itself does not resolve the inside/outside orientation ambiguity, we further propose to incorporate an objective function from Parametric Gauss Reconstruction (PGR). We propose to iteratively update normals by alternating between WNNC-based normal updates and PGR-based gradient descents, which leads to an embarrassingly simple yet effective iterative algorithm that allows fast and high-quality convergence to a globally consistent normal vector field. Furthermore, our proposed algorithm only involves repeatedly evaluating the winding number formula and its derivatives, which can be accelerated and parallelized using a treecode-based approximation algorithm due to their special structures. Exploiting this fact, we implement a GPU-accelerated treecode-based solver. Our GPU (and even CPU) implementation can be significantly faster than the recent state-of-the-art methods for normal orientation from raw points. Our code is integrated with the popular PyTorch framework to facilitate further research into winding numbers, and is publicly available at https://jsnln.github.io/wnnc/index.html.

Approximate bregman proximal gradient algorithm for relatively smooth nonconvex optimization

AbstractIn this paper, we propose the approximate Bregman proximal gradient algorithm (ABPG) for solving composite nonconvex optimization problems. ABPG employs a new distance that approximates the Bregman distance, making the subproblem of ABPG simpler to solve compared to existing Bregman-type algorithms. The subproblem of ABPG is often expressed in a closed form. Similarly to existing Bregman-type algorithms, ABPG does not require the global Lipschitz continuity for the gradient of the smooth part. Instead, assuming the smooth adaptable property, we establish the global subsequential convergence under standard assumptions. Additionally, assuming that the Kurdyka–Łojasiewicz property holds, we prove the global convergence for a special case. Our numerical experiments on the $$\ell _p$$ ℓ p regularized least squares problem, the $$\ell _p$$ ℓ p loss problem, and the nonnegative linear system show that ABPG outperforms existing algorithms especially when the gradient of the smooth part is not globally Lipschitz or even locally Lipschitz continuous.

Single- and Multi-Depot Optimization for UAV-Based IoT Data Collection in Neighborhoods

In this paper, we investigate the problem of deploying the minimum number of Unmanned Aerial Vehicles (UAVs) and determining their flying tours to collect data from all Internet of Things (IoT) sensors. We study this problem in a scenario with neighborhoods where a UAV can collect data from an IoT sensor if the distance between them is less than the wireless communication range of the IoT sensor. Since UAVs are powered by batteries with a limited amount of energy, we assume that the total energy consumed during the flying tour of each UAV is bounded by a given budget. We present the Minimum rooted drone Deployment Problem with Neighborhoods (MDPN), which is NP − hard , and propose two approximation algorithms for the single-depot case, where one of them is a bi-criteria approximation algorithm that returns a solution whose tour’s cost is violated by a factor of 1 + ϵ. Furthermore, we extend these two algorithms to the multi-depot scenario. Finally, we evaluate our algorithms in three different scenarios: the ideal one where the communication range is a circle and the data transfer rate is constant, and two more realistic scenarios where we introduce some degree of irregularity in the communication range and a non-constant rate in data transfer.

General relationship of local topologies, global dynamics, and bifurcation in cellular networks

Cellular networks realize their functions by integrating intricate information embedded within local structures such as regulatory paths and feedback loops. However, the precise mechanisms of how local topologies determine global network dynamics and induce bifurcations remain unidentified. A critical step in unraveling the integration is to identify the governing principles, which underlie the mechanisms of information flow. Here, we develop the cumulative linearized approximation (CLA) algorithm to address this issue. Based on perturbation analysis and network decomposition, we theoretically demonstrate how perturbations affect the equilibrium variations through the integration of all regulatory paths and how stability of the equilibria is determined by distinct feedback loops. Two illustrative examples, i.e., a three-variable bistable system and a more intricate epithelial-mesenchymal transition (EMT) network, are chosen to validate the feasibility of this approach. These results establish a solid foundation for understanding information flow across cellular networks, highlighting the critical roles of local topologies in determining global network dynamics and the emergence of bifurcations within these networks. This work introduces a novel framework for investigating the general relationship between local topologies and global dynamics of cellular networks under perturbations.