Approximation Factor Research Articles

Means of Hitting Times for Random Walks on Graphs: Connections, Computation, and Optimization

For random walks on graph \(\mathcal{G}\) with \(n\) vertices and \(m\) edges, the mean hitting time \(H_{j}\) from a vertex chosen from the stationary distribution to vertex \(j\) measures the importance for \(j\) , while the Kemeny constant \(\mathcal{K}\) is the mean hitting time from one vertex to another selected randomly according to the stationary distribution. In this paper, we first establish a connection between the two quantities, representing \(\mathcal{K}\) in terms of \(H_{j}\) for all vertices. We then develop an efficient algorithm estimating \(H_{j}\) for all vertices and \(\mathcal{K}\) in nearly linear time of \(m\) . Moreover, we extend the centrality \(H_{j}\) of a single vertex to \(H(S)\) of a vertex set \(S\) , and establish a link between \(H(S)\) and some other quantities. We further study the NP-hard problem of selecting a group \(S\) of \(k\ll n\) vertices with minimum \(H(S)\) , whose objective function is monotonic and supermodular. We finally propose two greedy algorithms approximately solving the problem. The former has an approximation factor \((1-\frac{k}{k-1}\frac{1}{e})\) and \(O(kn^{3})\) running time, while the latter returns a \((1-\frac{k}{k-1}\frac{1}{e}-\epsilon)\) -approximation solution in nearly-linear time of \(m\) , for any parameter \(0 \lt \epsilon \lt 1\) . Extensive experiment results validate the performance of our algorithms.

Computing Prominent Skyline on Massive Data

AbstractIn many practical applications, skyline query is an important operation to return the pareto optimal tuples, which provides a candidate set for the optimum. On massive data, skyline often reports too many results, the users will be overwhelmed and be difficult to find the desired information easily. This paper devises P-skyline to reduce the size of the returned results. Given the approximation factor, P-skyline only generates the prominent skyline results by the definition of p-dominance. To the best of our knowledge, this paper is the first work to study P-skyline problem. This paper first proposes a baseline algorithm, which requires one full table scan to compute the results. It is found that baseline algorithm incurs a relatively high execution cost on massive data. Then, PSTP algorithm is proposed, which consists of two stages: candidate acquisition and refinement. On the presorted table, PSTP utilizes selective retrieval and selective checking to process P-skyline with much lower I/O cost and computation cost. The extensive experimental results, conducted on synthetic and real-life data sets, show that PSTP can compute P-skyline on massive data efficiently.

On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.

CAUSALdb2: an updated database for causal variants of complex traits.

Unraveling the causal variants from genome wide association studies (GWASs) is pivotal for understanding genetic underpinnings of complex traits and diseases. Despite continuous efforts, tools to refine and prioritize GWAS signals need enhancement to address the direct causal implications of genetic variations. To overcome challenges related to statistical fine-mapping in identifying causal variants, CAUSALdb has been updated with novel features and comprehensive datasets, morphing into CAUSALdb2. This expanded repository integrates 15057 updated GWAS summary statistics across 10839 unique traits and implements both LD-based and LD-free fine-mapping approaches, including innovative applications of approximate Bayes Factor and SuSiE. Additionally, by incorporating larger LD reference panels such as TOPMED and UK Biobank, and integrating functional annotations via PolyFun, CAUSALdb2 enhances the accuracy and context of fine-mapping results. The database now supports interrogation of additional causal signals and offers sophisticated visualizations to aid researchers in deciphering complex genetic architectures. By facilitating a deeper and more precise characterisation of causal variants, CAUSALdb2 serves as a crucial tool for advancing the genetic analysis of complex diseases. Available freely, CAUSALdb2 continues to set benchmarks in the post-GWAS era, fostering the development of targeted diagnostics and therapeutics derived from responsible genetic research. Explore these advancements at http://mulinlab.org/causaldb.

On MAX–SAT with cardinality constraint

We consider the weighted MAX–SAT problem with an additional constraint that at mostk variables can be set to true. We call this problem k–WMAX–SAT. This problem admits a (1−1e)-factor approximation algorithm in polynomial time [Sviridenko, Algorithmica (2001)] and it is proved that there is no (1−1e+ϵ)-factor approximation algorithm in f(k)⋅no(k) time for Maximum Coverage, the unweighted monotone version of k–WMAX–SAT [Manurangsi, SODA 2020]. Therefore, we study two restricted versions of the problem in the realm of parameterized complexity.1.When the input is an unweighted 2–CNF formula (the problem is called k–MAX–2SAT), we design an efficient polynomial-size approximate kernelization scheme. That is, we design a polynomial-time algorithm that given a 2–CNF formula ψ and ϵ>0, compresses the input instance to a 2–CNF formula ψ⋆ such that any c-approximate solution of ψ⋆ can be converted to a c(1−ϵ)-approximate solution of ψ in polynomial time.2.When the input is a planar CNF formula, i.e., the variable-clause incidence graph is a planar graph, we show the following results:•There is an FPT algorithm for k–WMAX–SAT on planar CNF formulas that runs in 2O(k)⋅(C+V) time.•There is a polynomial-time approximation scheme for k–WMAX–SAT on planar CNF formulas that runs in time 2O(1ϵ)⋅k⋅(C+V). The above-mentioned C and V are the number of clauses and variables of the input formula respectively.

Linear regression model of structured column packing form factor

Semi-empirical correlations are commonly used to predict packed column pressure drop. As a downside, these models rely on packing-specific parameters that are obtained through laborious experiments.This work aims to eliminate packing-specific parameters for modeling pressure drop of structured packings. A multiple linear regression model is used to predict the packing form factor directly from packing properties. The form factor is then used with the model of Maćkowiak (2010) to calculate pressure drop. This tandem model approach therefore allows for pressure drop predictions directly from packing properties.The regression model includes model features to describe packing properties, such as packing geometry, perforations, packing material or packing surface. Model coefficients are determined using experimental pressure drop results of 28 different structured column packings manufactured from metal, plastic or ceramic. The fitted regression model yields very good (mean absolute error 1.86%) approximation of packing form factor. Model feature importance analysis suggests packing porosity, packing angle and packing perforations exert the greatest influence on the form factor.Leave-one-out cross-validation is performed to evaluate predictive model performance beyond training data. Results indicate the model’s reliable predictions of the form factor (mean absolute error 3.1%) and strong agreement between pressure drop predictions and experimental data (mean absolute error 23.4%).

Parameterized inapproximability of Morse matching

We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of 2log(1−ϵ)⁡n. Our second result shows that Min-Morse Matching is W[P]-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor ρ. The above hardness results are applicable to complexes of dimension ≥2.On the positive side, we provide a factor O(nlog⁡n) approximation algorithm for Min-Morse Matching on 2-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa–Farber model of random complexes.

Stress Field near Crack Tips in Pressurized Thick-Walled Cylinders with Single and Multiple Cracks in Linear Elastic Fracture Mechanics: Analytical and Numerical Analysis of Stress Field and Crack Behavior

<div>Arrays of radial cracks often appear at the bore of pressurized cylinders, posing potential safety risks and leading to possible structural failures. This article presents an analytical approach to evaluate the stress field arising from single or multiple uniform radial cracks in thick-walled pressurized cylinders within the context of linear elastic fracture mechanics (LEFM) under mode-I loading. This formulation is based on the fundamental equations of elasticity and approximations of stress intensity factors (SIF) reported in the literature. Hence, the SIF were revisited and their range of validity was highlighted. The study considers two types of internal pressure loading: one applied only to the cylinder’s inner surface with no pressure on the crack faces and another applied to both the inner surface and the crack faces. The influence of the number and length of cracks relative to cylinder thickness on the stress field is analyzed. A finite element model of the pressurized vessel is constructed to verify the proposed analytical solution, showing good agreement with numerical results for a limited number of cracks.</div>

Simulation of gas mixture dynamics in a pipeline network using explicit staggered-grid discretization

We develop an explicit staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge boundary values that are more complex than for a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

The analysis of Hermite factor of BKZ algorithm on small lattices

Lattice cryptography is one of the promising directions in modern cryptography research. Digital signatures and key encapsulation mechanisms on lattices have already been used in practice. In the future, such quantum-resistant transformations on lattices replace all standards that are not resistant to attacks on quantum computers. This makes the analysis of their security extremely relevant. Analysis of the security of cryptographic transformations on lattices is often reduced to the estimation of the minimum block size in the lattice reduction algorithm. For the expansion of small vectors, a reduction algorithm can be obtained for a given block size, the GSA model is often used, which uses the so-called Hermitian factor to predict the size of the vectors that the lattice reduction algorithm can obtain given the parameters. Asymptotic formulas have been developed to evaluate it in practice, but the question of their accuracy on cryptographic lattices has not been fully investigated. The work obtained estimates of the accuracy of the existing asymptotic estimates of the Hermite factor for lattices of sizes 120, 145, 170 for the classical BKZ algorithm. Research was conducted using the fpylll library. It was shown that the existing estimators are equivalent from a practical point of view and have a sufficiently small root mean square deviation from the true values. A formula was obtained that binds the root-mean-square error of approximation of the Hermit factor to the cryptographic parameters of lattices. The obtained results are useful for refining the security assessments of existing cryptographic transformations.

Multivariate Stochastic Volatility with Co-Heteroscedasticity

Abstract A new methodology that decomposes shocks into homoscedastic and heteroscedastic components is developed. This specification implies there exist linear combinations of heteroscedastic variables that eliminate heteroscedasticity; a property known as co-heteroscedasticity. The heteroscedastic part of the model uses a multivariate stochastic volatility inverse Wishart process. The resulting model is invariant to the ordering of the variables, which is shown to be important for volatility estimation. By incorporating testable co-heteroscedasticity restrictions, the specification allows estimation in moderately high-dimensions. The computational strategy uses a novel particle filter algorithm, a reparameterization that substantially improves algorithmic convergence and an alternating-order particle Gibbs that reduces the amount of particles needed for accurate estimation. An empirical application to a large Vector Autoregression (VAR) is provided, finding strong evidence for co-heteroscedasticity and that the new method outperforms some previously proposed methods in terms of forecasting at all horizons. It is also found that the structural monetary shock is 98.8 % homoscedastic, and that investment and the SP 500 index are nearly 100 % determined by fat tail heteroscedastic shocks. A Monte Carlo experiment illustrates that the new method estimates well the characteristics of approximate factor models with heteroscedastic errors.

Connected k-Center and k-Diameter Clustering

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. The former problem has been introduced by Ge et al. (ACM Trans Knowl Discov Data 2(2):1–35, 2008. https://doi.org/10.1145/1376815.1376816) to model clustering of data sets with both attribute and relationship data. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graphG on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(log2k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log ^2k)$$\\end{document}-approximation algorithm for the disjoint connected k-center and k-diameter problem. For Euclidean spaces of constant dimension and for metrics with constant doubling dimension, the approximation factor improves to O(1). Our algorithm works by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Many-Body Entropies and Entanglement from Polynomially Many Local Measurements

Estimating global properties of many-body quantum systems such as entropy or bipartite entanglement is a notoriously difficult task, typically requiring a number of measurements or classical postprocessing resources growing exponentially in the system size. In this work, we address the problem of estimating global entropies and mixed-state entanglement via partial-transposed (PT) moments and show that efficient estimation strategies exist under the assumption that all the spatial correlation lengths are finite. Focusing on one-dimensional systems, we identify a set of approximate factorization conditions (AFCs) on the system density matrix, which allow us to reconstruct entropies and PT moments from information on local subsystems. This identification yields a simple and efficient strategy for entropy and entanglement estimation. Our method could be implemented in different ways, depending on how information on local subsystems is extracted. Focusing on randomized measurements providing a practical and common measurement scheme, we prove that our protocol requires only polynomially many measurements and postprocessing operations, assuming that the state to be measured satisfies the AFCs. We prove that the AFCs hold for finite-depth quantum-circuit states and translation-invariant matrix-product density operators and provide numerical evidence that they are satisfied in more general, physically interesting cases, including thermal states of local Hamiltonians. We argue that our method could be practically useful to detect bipartite mixed-state entanglement for large numbers of qubits available in today’s quantum platforms. Published by the American Physical Society 2024

Adaptive Shivers Sort: An Alternative Sorting Algorithm

We present a new sorting algorithm, called adaptive ShiversSort , that exploits the existence of monotonic runs for sorting efficiently partially sorted data. This algorithm is a variant of the well-known algorithm TimSort , which is the sorting algorithm used in standard libraries of programming languages, such as Python or Java (for non-primitive types). More precisely, adaptive ShiversSort is a so-called \(k\) -aware merge-sort algorithm, a class that captures ‘ TimSort -like’ algorithms and that was introduced by Buss and Knop. In this article, we prove that, although adaptive ShiversSort is simple to implement and differs only slightly from TimSort , its computational cost, in number of comparisons performed, is optimal within the class of natural merge-sort algorithms, up to a small additive linear term. This makes adaptive ShiversSort the first \(k\) -aware algorithm to benefit from this property, which is also a 33% improvement over TimSort 's worst-case. This suggests that adaptive ShiversSort could be a strong contender for being used instead of TimSort . Then, we investigate the optimality of \(k\) -aware algorithms. We give lower and upper bounds on the best approximation factors of such algorithms, compared to optimal stable natural merge-sort algorithms. In particular, we design generalisations of adaptive ShiversSort whose computational costs are optimal up to arbitrarily small multiplicative factors.

A better LP rounding for feedback arc set on tournaments

We present a randomized algorithm to approximate the feedback arc set problem on weighted tournaments, a classic well-studied NP-hard problem. Our algorithm is based on rounding its standard linear programming relaxation. It improves the previously best-known LP rounding algorithm by achieving an approximation factor of 2.127 (best known was 2.5). As a result, we have found a better upper bound for the integrality gap of the corresponding LP.

A Sparse Approximate Factor Model for High-Dimensional Covariance Matrix Estimation and Portfolio Selection

Abstract We propose a novel estimation approach for the covariance matrix based on the l1-regularized approximate factor model (AFM). Our sparse approximate factor (SAF) covariance estimator allows for the existence of weak factors and hence relaxes the pervasiveness assumption generally adopted for the standard AFM. We prove the consistency of the covariance matrix estimator under the Frobenius norm as well as the consistency of the factor loadings and the factors. Our Monte Carlo simulations reveal that the SAF covariance estimator has superior properties in finite samples for low and high dimensions and different designs of the covariance matrix. Moreover, in an out-of-sample portfolio forecasting application, the estimator uniformly outperforms alternative portfolio strategies based on alternative covariance estimation approaches and modeling strategies including the 1/N-strategy.

An approximate block factorization preconditioner for mixed-dimensional beam-solid interaction

This paper presents a scalable approximate block factorization preconditioner for mixed-dimensional models in beam-solid interaction and their application in engineering. In particular, it studies the linear systems arising from a regularized mortar-type approach for embedding geometrically exact beams into solid continua. Due to the lack of block diagonal dominance of the arising 2 × 2 block system, an approximate Block-LU preconditioner is used. It exploits the sparsity structure of the beam sub-block to construct a sparse approximate inverse, which is then not only used to explicitly form an approximation of the Schur complement, but also acts as a smoother within the prediction and correction step of the arising Block-LU preconditioner. The Schur complement equation is tackled with an algebraic multigrid method. Although, for now, the beam sub-block is tackled by a one-level method only, the multi-level nature of the computationally demanding Schur equation delivers a scalable preconditioner in practice. In numerical test cases, the influence of different algorithmic parameters on the quality of the sparse approximate inverse is studied and the weak scaling behavior of the proposed preconditioner on up to 1000 MPI ranks is demonstrated. In addition, the robustness of the proposed method regarding material parameters and geometric properties is shown, before the preconditioner is finally applied for the analysis of steel-reinforced concrete structures in civil engineering.

Approximate settlement influence factors for bucket foundations

ABSTRACT The response of bucket foundations for offshore structures has received significant attention from researchers but most of the reported studies focused on ultimate capacity. In this study, a comprehensive series of finite element analyses have been performed to investigate directly how the skirt geometry affects the elastic settlement of bucket foundations of arbitrary rigidity embedded in either a homogeneous or heterogeneous soil medium with finite layer thickness. The effect of soil drainage conditions is also explored. The results are graphically presented in the familiar form of settlement influence factors which can be used in hand calculations to predict the settlement of bucket foundations for a wide range of practical cases. Approximating expressions that capture the settlement influence factors well are developed. A worked example is given to illustrate the use of the graphical and approximating solutions.

Selecting the number of factors in approximate factor models using group variable regularization

. We propose a novel method for the estimation of the number of factors in approximate factor models. The model is based on a penalized maximum likelihood approach incorporating an adaptive hierarchical lasso penalty function that enables setting entire columns of the factor loadings matrix to zero, which corresponds to removing uninformative factors. Additionally, the model is capable of estimating weak factors by allowing for sparsity in the non zero columns. We prove that the proposed estimator consistently estimates the true number of factors. Simulation experiments reveal superior selection accuracies for our method in finite samples over existing approaches, especially for data with cross-sectional and serial correlations. In an empirical application on a large macroeconomic dataset, we show that the mean squared forecast errors of an approximate factor model are lower if the number of included factors is selected by our method.

Geometry and temperature effects on tensile properties and failure behaviors of open-hole and bolted-joint CF/PEKK composites

Temperature-dependent tensile properties of carbon-fiber-reinforced polyetherketoneketone (CF/PEKK) composites with varying width-to-diameter (W/D) ratios in open-hole (OH) and bolted joint (BJ) were investigated. CF/PEKK exhibited higher modulus and strength at lower temperature due to restricted polymer chain mobility, and lower values at elevated temperature due to increased polymer ductility. OH net tensile strength decreased with decreasing W/D, with strain retention dropping significantly at W/D ≤ 2. The approximate average strain concentration factor determined from digital image correlation slightly exceeded theoretical values due to semi-crystalline nature of PEKK. Distinct failure behaviors highlighting the complex interplay between temperature, W/D, and failure characteristics. Evaluating W/D and temperature effects on BJ CF/PEKK revealed changing failure modes. Bolted joint efficiency emphasizing the need to optimize W/D ratios above 2 to achieve high efficiency. These findings highlight the geometry and temperature interplay affects CF/PEKK composites, which are crucial for designing high-performance materials, especially for aerospace applications.