Approximation Algorithm Research Articles

Joint optimization of application placement and resource allocation for enhanced performance in heterogeneous multi-server systems

Efficiently placing applications remains a critical challenge across diverse multi-server environments, including web hosting centers, cloud computing, and edge computing environments. Unfortunately, most existing studies tend to overlook the crucial aspect of resource allocation, leading to suboptimal system performance. To address this gap, there is a pressing need to holistically explore both application placement and resource allocation in a unified manner. In this paper, we introduce the place and allocate problem in heterogeneous multi-server systems, a novel approach aiming at simultaneously optimizing the placement and allocation of applications to maximize the overall utility of the heterogeneous multi-server system. Our proposed methodology harnesses the interplay between application placement and resource allocation, showcasing substantial improvements in system utility. To model individual application performance concerning their allocated resources, we employ utility functions. For concave utility functions, we present an approximation algorithm that operates efficiently with a time complexity of O(mn3(logC)2), where n represents the number of applications, m is the number of servers, and C denotes the maximum available resource capacity of each server. Furthermore, we extend our approach to accommodate more general scenarios that involve applications with nonconcave utility functions and using multiple types of resources. Our study includes comprehensive experimental evaluations conducted on applications with both synthetic and real-world utility functions. Results consistently showcase that our algorithms achieve over 96.9% of optimal performance on average. Additionally, comparative analysis against several practical heuristics reveal that our algorithms outperform these methods by up to 4.3 times in total utility.

A novel combination prediction model of ultra-short-term wind speed

The accurate prediction of ultra-short-term wind speed has important theoretical significance and practical application value. In this paper, a combination prediction model of ultra-short-term wind speed based on variational mode decomposition is proposed. Firstly, the variational mode decomposition algorithm is introduced to decompose the ultra-short-term wind speed and obtain the components of different frequency. The approximate entropy algorithm is used to calculate the complexity of each component. According to the calculation results, echo state network is selected to predict high complexity components, support vector machine is used to predict medium complexity components, and autoregressive integrated moving average model is used to predict low complexity components. Then, the predicted values of each component are added to get the final prediction result. Gray wolf algorithm is used to optimize the model parameters of support vector machine and echo state network. In addition, the approximate entropy calculation results show that compared with the original ultra-short-term wind speed, the complexity of the components obtained by the variational mode decomposition algorithm is reduced, which makes the modeling of the ultra-short-term wind speed system simpler. Finally, the validity of the model is verified by taking the ultra-short-term wind speed data of 5 minutes and 10 minutes sampling period actually collected as the research object. The results show that the prediction model proposed in this paper has better performance than other single or combination prediction models.

Synthetic lethality and the minimal genome size problem.

Gene knockout studies suggest that ~300 genes in a bacterial genome and ~1,100 genes in a yeast genome cannot be deleted without loss of viability. These single-gene knockout experiments do not account for negative genetic interactions, when two or more genes can each be deleted without effect, but their joint deletion is lethal. Thus, large-scale single-gene deletion studies underestimate the size of a minimal gene set compatible with cell survival. In yeast Saccharomyces cerevisiae, the viability of all possible deletions of gene pairs (2-tuples), and of some deletions of gene triplets (3-tuples), has been experimentally tested. To estimate the size of a yeast minimal genome from that data, we first established that finding the size of a minimal gene set is equivalent to finding the minimum vertex cover in the lethality (hyper)graph, where the vertices are genes and (hyper)edges connect k-tuples of genes whose joint deletion is lethal. Using the Lovász-Johnson-Chvatal greedy approximation algorithm, we computed the minimum vertex cover of the synthetic-lethal 2-tuples graph to be 1,723 genes. We next simulated the genetic interactions in 3-tuples, extrapolating from the existing triplet sample, and again estimated minimum vertex covers. The size of a minimal gene set in yeast rapidly approaches the size of the entire genome even when considering only synthetic lethalities in k-tuples with small k. In contrast, several studies reported successful experimental reductions of yeast and bacterial genomes by simultaneous deletions of hundreds of genes, without eliciting synthetic lethality. We discuss possible reasons for this apparent contradiction.IMPORTANCEHow can we estimate the smallest number of genes sufficient for a unicellular organism to survive on a rich medium? One approach is to remove genes one at a time and count how many of such deletion strains are unable to grow. However, the single-gene knockout data are insufficient, because joint gene deletions may result in negative genetic interactions, also known as synthetic lethality. We used a technique from graph theory to estimate the size of minimal yeast genome from partial data on synthetic lethality. The number of potential synthetic lethal interactions grows very fast when multiple genes are deleted, revealing a paradoxical contrast with the experimental reductions of yeast genome by ~100 genes, and of bacterial genomes by several hundreds of genes.

Polynomial Integrality Gap of Flow LP for Directed Steiner Tree

In the Directed Steiner Tree (DST) problem, we are given a directed graph \(G=(V,E)\) on \(n\) vertices with edge-costs \(c\in{\mathbb{R}}_{\geq 0}^{E}\) , a root vertex \(r\in V\) , and a set \(K\subseteq V\setminus\{r\}\) of \(k\) terminals. The goal is to find a minimum-cost subgraph of \(G\) that contains a path from \(r\) to every terminal \(t\in K\) . DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of \(O(k^{\epsilon})\) for any constant \(\epsilon {\gt} 0\) , and the best quasi-polynomial-time approximation ratio of \(O\left(\frac{\log^{2}k}{\log\log k}\right)\) . Towards understanding this gap, we study the integrality gap of the standard flow linear programming relaxation for the problem. We show that the linear program (LP) has an integrality gap of \(\Omega(n^{0.0418})\) . Previously, the integrality gap of the LP is only known to be \(\Omega\left(\frac{\log^{2}n}{\log\log n}\right)\) [Halperin et al., SODA’03 & SIAM J. Comput.] and \(\Omega(\sqrt{k})\) [Zosin-Khuller, SODA’02] in some instance with \(\sqrt{k}=O\left(\frac{\log n}{\log\log n}\right)\) . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in \(n\) , the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

An approximation algorithm for multiobjective mixed-integer convex optimization

AbstractIn this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

An extended Vogel’s approximation algorithm for efficiently solving Fermatean fuzzy solid transportation problems

An extended Vogel’s approximation algorithm for efficiently solving Fermatean fuzzy solid transportation problems

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

A Non-Metallic pipeline leak size recognition method based on CWT acoustic image transformation and CNN

Accurately identifying the pipeline leak size is crucial for risk assessment and timely rescue. In this study, a Convolutional Neural Network (CNN) based on Continuous Wavelet Transform (CWT) acoustic image transformation is proposed to identify small-sized leak in non-metallic pipes. Firstly, one-dimensional acoustic signals are filtered using the Piecewise Aggregate Approximation (PAA) algorithm to reduce noise and storage resource consumption. Then, the filtered signals are transformed into two-dimensional images by CWT to enrich signal feature information, serving as the input for the CNN. Further, a leak size recognition model based on CWT-CNN is established. The effectiveness of this model is verified using experimental data from a non-metallic pipeline leak test. A comparative analysis is conducted on diverse acoustic image transformation methods, including CWT, Gramian Angular Summation Field (GASF), and Relative Position Matrix (RPM). The results demonstrate the superiority of the CWT-CNN model in pipeline leak size recognition. Finally, the impact of the signal length in an acoustic image on recognition accuracy is also examined. The results demonstrate that when the signal length in an acoustic image is 0.75 s, the accuracy obtained by CWT-CNN can reach 95 %.

Enhancing the rationale of convolutional neural networks for glitch classification in gravitational wave detectors: a visual explanation

In the pursuit of detecting gravitational waves, ground-based interferometers (e.g. LIGO, Virgo, and KAGRA) face a significant challenge: achieving the extremely high sensitivity required to detect fluctuations at distances significantly smaller than the diameter of an atomic nucleus. Cutting-edge materials and innovative engineering techniques have been employed to enhance the stability and precision of the interferometer apparatus over the years. These efforts are crucial for reducing the noise that masks the subtle gravitational wave signals. Various sources of interference, such as seismic activity, thermal fluctuations, and other environmental factors, contribute to the total noise spectra characteristic of the detector. Therefore, addressing these sources is essential to enhance the interferometer apparatus’s stability and precision. Recent research has emphasised the importance of classifying non-stationary and non-Gaussian glitches, employing sophisticated algorithms and machine learning methods to distinguish genuine gravitational wave signals from instrumental artefacts. The time-frequency-amplitude representation of these transient disturbances exhibits a wide range of new shapes, variability, and features, reflecting the evolution of interferometer technology. In this study, we developed a convolutional neural network model to classify glitches using spectrogram images from the Gravity Spy O1 dataset. We employed score-class activation mapping and the uniform manifold approximation and projection algorithm to visualise and understand the classification decisions made by our model. We assessed the model’s validity and investigated the causes of misclassification from these results.

Power-Efficient Resource Allocation for Active STAR-RIS-Aided SWIPT Communication Systems

Simultaneous wireless information and power transfer (SWIPT) has emerged as a pivotal technology in 6G, offering an efficient means of delivering energy to a large quantity of low-power devices while transmitting data concurrently. To address the challenges of obstructions, high path loss, and significant energy consumption associated with long-distance communication, this work introduces a novel alternating iterative optimization strategy. The proposed approach combines active simultaneous transmission and reflection of reconfigurable intelligent surfaces (STAR-RIS) with SWIPT to maximize spectrum efficiency and reduce overall system energy consumption. This method addresses the considerable energy demands inherent in SWIPT systems by focusing on reducing the power output from the base station (BS) while meeting key constraints: the communication rate for information receivers (IRs) and minimum energy levels for energy receivers (ERs). Given complex interactions between variables, the solution involves an alternating iterative optimization process. In the first stage of this approach, the passive beamforming variables are kept constant, enabling the use of semi-definite relaxation (SDR) and successive convex approximation (SCA) algorithms to optimize active beamforming variables. In the next stage, with active beamforming variables fixed, penalty-based algorithms are applied to fine-tune the passive beamforming variables. This iterative process continues, alternating between active and passive beamforming optimization, until the system converges on a stable solution. The simulation results indicated that the proposed system configuration, which leverages active STAR-RIS, achieves lower energy consumption and demonstrates improved performance compared to configurations utilizing passive RIS, active RIS, and passive STAR-RIS. This evidence suggests that the proposed approach can significantly contribute to advancing energy efficiency in 6G systems.

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.

Max-flow min-cut theorem in quantum computing

The max-flow min-cut theorem in graph theory states that the maximum flow of a network is equal to the minimum cut of edges of the flow network. Max-flow and min-cut are related as two primal–dual linear programs. The theorem is applicable in applications like network connectivity, graph matching problems, transportation and logistics, and scheduling problems. The aim of the paper is to model the classical max-flow and min-cut theorem in quantum computing. Two well-known methods for quantum optimization are quantum annealing and quantum approximate optimization algorithm. The max-flow min-cut is converted to its equivalent model of the above two methods for execution in polynomial time. This paper shows detailed classical to quantum conversion, analysis, implementation, and result of the theorem.

SuperTAD-Fast: Accelerating Topologically Associating Domains Detection Through Discretization.

High-throughput chromosome conformation capture (Hi-C) technology captures spatial interactions of DNA sequences into matrices, and software tools are developed to identify topologically associating domains (TADs) from the Hi-C matrices. With structural information theory, SuperTAD adopted a dynamic programming approach to find the TAD hierarchy with minimal structural entropy. However, the algorithm suffers from high time complexity. To accelerate this algorithm, we design and implement an approximation algorithm with a theoretical performance guarantee. We implemented a package, SuperTAD-Fast. Using Hi-C matrices and simulated data, we demonstrated that SuperTAD-Fast achieved great runtime improvement compared with SuperTAD. SuperTAD-Fast shows high consistency and significant enrichment of structural proteins from Hi-C data of human cell lines in comparison with the existing six hierarchical TADs detecting methods.

Generating information-dense promoter sequences with optimal string packing.

Dense arrangements of binding sites within nucleotide sequences can collectively influence downstream transcription rates or initiate biomolecular interactions. For example, natural promoter regions can harbor many overlapping transcription factor binding sites that influence the rate of transcription initiation. Despite the prevalence of overlapping binding sites in nature, rapid design of nucleotide sequences with many overlapping sites remains a challenge. Here, we show that this is an NP-hard problem, coined here as the nucleotide String Packing Problem (SPP). We then introduce a computational technique that efficiently assembles sets of DNA-protein binding sites into dense, contiguous stretches of double-stranded DNA. For the efficient design of nucleotide sequences spanning hundreds of base pairs, we reduce the SPP to an Orienteering Problem with integer distances, and then leverage modern integer linear programming solvers. Our method optimally packs sets of 20-100 binding sites into dense nucleotide arrays of 50-300 base pairs in 0.05-10 seconds. Unlike approximation algorithms or meta-heuristics, our approach finds provably optimal solutions. We demonstrate how our method can generate large sets of diverse sequences suitable for library generation, where the frequency of binding site usage across the returned sequences can be controlled by modulating the objective function. As an example, we then show how adding additional constraints, like the inclusion of sequence elements with fixed positions, allows for the design of bacterial promoters. The nucleotide string packing approach we present can accelerate the design of sequences with complex DNA-protein interactions. When used in combination with synthesis and high-throughput screening, this design strategy could help interrogate how complex binding site arrangements impact either gene expression or biomolecular mechanisms in varied cellular contexts.

Power system fault diagnosis with quantum computing and efficient gate decomposition

Power system fault diagnosis is crucial for identifying the location and causes of faults and providing decision-making support for power dispatchers. However, most classical methods suffer from significant time-consuming, memory overhead, and computational complexity issues as the scale of the power system concerned increases. With rapid development of quantum computing technology, the combinatorial optimization method based on quantum computing has shown certain advantages in computational time over existing methods. Given this background, this paper proposes a quantum computing based power system fault diagnosis method with the quantum approximate optimization algorithm. The proposed method reformulates the fault diagnosis problem as a Hamiltonian by using Ising model, which completely preserves the coupling relationship between faulty components and various operations of protective relays and circuit breakers. Additionally, to enhance problem-solving efficiency under current equipment limitations, the symmetric equivalent decomposition method of multi-z-rotation gate is utilized. Furthermore, the small probability characteristics of power system events is utilized to reduce the number of qubits. Simulation results based on the test system show that the proposed methods can achieve the same optimal results with a faster speed compared with the classical higher-order solver provided by D-Wave.

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2 . Specifically, we obtain a \(1+\frac{1}{\sqrt {2}}+\epsilon \approx 1.707+\epsilon \) approximation in bipartite graphs and a 1.973 + ϵ approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms’ approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS’21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

Approximation algorithms for solving the vertex-traversing-constrained mixed Chinese postman problem

Approximation algorithms for solving the vertex-traversing-constrained mixed Chinese postman problem

Improved approximation for two-dimensional vector multiple knapsack

We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.Our main result is a (1−ln⁡22−ε)-approximation algorithm for 2VMK, for every fixed ε>0, thus improving the best known ratio of (1−1e−ε) which follows as a special case from a result of Fleischer et al. (2011) [6].Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to ≈m⋅ln⁡2≈0.693⋅m of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

A new heuristic for the continuous fixed-radius disc cover problem

The continuous fixed-radius disc cover (CFDC) problem is a location model to cover the given demand points by finding the minimum number of fixed-radius discs in the plane. It is an NP-hard problem, and only a few approximate algorithms have been proposed over the past three decades. In the article, a new heuristic called dynamic reduction (DR) is introduced. The heuristic works by initiating an upper bound on the number of discs, and then reducing the number by removing redundant discs during updates to the disc centres. Experimental results, using different radius settings on some datasets of a travelling salesman problem library (TSP-Lib), show that the proposed heuristic greatly outperforms the recent approximation algorithm in accuracy. Notably, an enhancement of the heuristic is also provided.

A High-Precision Sub-Grid Parameterization Scheme for Clear-Sky Direct Solar Radiation in Complex Terrain—Part I: A High-Precision Fast Terrain Occlusion Algorithm

In atmospheric modeling, sub-grid parameterization is an important method for studying the topographic effects of solar radiation using high-resolution Digital Elevation Model (DEM) data. For reducing the amount of computation, some approximate methods that can lead to errors are used in existing sub-grid parameterization schemes for clear-sky direct solar radiation (SPS-CSDSR). The lack of a high-precision fast terrain occlusion algorithm (HPFTOA) remains one of the biggest constraints in this field. This study proposed an HPFTOA. It mainly uses two kinds of acceleration algorithms. One method is to use a dynamic, lossless, and fast occlusion search radius. Another way is to use the rectangular grid for calculations within the accuracy of DEM data to avoid coordinate projection conversions. The test results indicate that the HPFTOA can carry out large-scale computation based on DEM data with a resolution of 90 m. Because it rarely uses approximation algorithms and considers the curvature of the Earth, SPS-CSDSR can achieve unprecedented precision. The HPFTOA can also be used in the fields of mountain solar energy assessment, remote sensing, and telemetry, including terrain-obscuring the probe. As computer performance improves and algorithms and execution code are optimized, the application prospects will be very broad.