Approximation Algorithm Research Articles

Enhancing the Rationale of Convolutional Neural Networks for Glitch Classification in Gravitational Wave Detectors: A Visual Explanation

Abstract 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.

Algorithms for Densest Subgraphs of Vertex-Weighted Graphs

Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding a subgraph with maximum mean density. However, most densest subgraph-finding algorithms are based on edge-weighted graphs, where edge weights can only represent a single value dimension, whereas practical applications involve multiple dimensions. To resolve the challenge, we propose two algorithms for resolving the densest subgraph problem in a vertex-weighted graph. First, we present an exact algorithm that builds upon Goldberg’s original algorithm. Through theoretical exploration and analysis, we rigorously verify our proposed algorithm’s correctness and confirm that it can efficiently run in polynomial time O(n(n + m)log2n) is its temporal complexity. Our approach can be applied to identify closely related subgroups demonstrating the maximum average density in real-life situations. Additionally, we consistently offer an approximation algorithm that guarantees an accurate approximation ratio of 2. In conclusion, our contributions enrich theoretical foundations for addressing the densest subgraph problem.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Hydrothermal Dissolution of Reactive Silica Materials in Sodium Hydroxide Lyes

AbstractReactive silica materials like vitreous silica or cristobalite were dissolved at temperatures between 100 and 220 °C for up to 7 days in NaOH lyes. The liquid and solid reaction products were analyzed chemically. Sodium water glasses with silica concentrations of up to 27 wt % and molar SiO2/Na2O ratios of up to 3.7 were obtained. The experimental data were evaluated by an approximation algorithm. The initial size reduction rates ranged between 0.1 and 7 µm h−1, depending on the temperature and the choice of starting materials. Correlations between thermodynamics and kinetics were discussed.

Structural iterative rounding for generalized k-median problems

AbstractThis paper considers approximation algorithms for generalized k-median problems. These problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for k-median with outliers as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018). The main technical innovation is allowing richer constraint sets in the iterative rounding and using the structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios $$6.994 + \epsilon $$ 6.994 + ϵ and $$6.387 + \epsilon $$ 6.387 + ϵ for k-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio $$7.081 + \epsilon $$ 7.081 + ϵ for both problems as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018).

Distributed Independent Sets in Interval and Segment Intersection Graphs

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Hardness and Approximability of Dimension Reduction on the Probability Simplex

Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like engineering, biology, astronomy, and economics. In this paper, we consider the following dimensionality reduction instance: Given an n-dimensional probability distribution p and an integer m<n, we aim to find the m-dimensional probability distribution q that is the closest to p, using the Kullback–Leibler divergence as the measure of closeness. We prove that the problem is strongly NP-hard, and we present an approximation algorithm for it.

On the total energy consumption of scalable cache-aided multi-CPU cell-free massive MIMO systems

In conventional cell-free (CF) systems, the scalability and feasibility are hindered by the fact that a single central processing unit (CPU) manages all access points and the stringent synchronization requirement for coherent transmission (CT) across the entire network. Multi-CPU CF systems emerge as a scalable and feasible alternative, yet they still encounter the typical CF system challenge of high downlink transmit energy consumption from extensive fronthaul and backhaul transmissions. To tackle this issue, we propose a scalable downlink partial coherent transmission (PCT) strategy for multi-CPU CF systems to provide high spectral efficiency (SE), reduce transmission time and thereby lower energy consumption, while keeping decoding complexity manageable. Based on the PCT strategy, we introduce scalable cache-aided multi-CPU CF systems and develop the corresponding total energy consumption (TEC) model. A successive convex approximation algorithm is proposed to obtain the suboptimal cache placement to minimize the TEC. Simulation results indicate that the proposed PCT-based scalable cache-aided multi-CPU CF systems can significantly reduce the TEC and consistently maintain the TEC at a relatively low level across all Zipf parameters by leveraging the high SE of CT and the independent transmission of non-coherent transmission.

Hardness and Approximability of Dimension Reduction on the Probability Simplex

Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like engineering, biology, astronomy, and economics. In this paper, we consider the following dimensionality reduction instance: Given an n-dimensional probability distribution p and an integer m<n, we aim to find the m-dimensional probability distribution q that is the closest to p, using the Kullback–Leibler divergence as the measure of closeness. We prove that the problem is strongly NP-hard, and we present an approximation algorithm for it.

Hybrid GRU-CNN bilinear parameters initialization for quantum approximate optimization algorithm

The Quantum Approximate Optimization Algorithm (QAOA), a pivotal paradigm in the realm of variational quantum algorithms (VQAs), offers promising computational advantages for tackling combinatorial optimization problems. Well-defined initial circuit parameters, responsible for preparing a parameterized quantum state encoding the solution, play a key role in optimizing QAOA. However, classical optimization techniques encounter challenges in discerning optimal parameters that align with the optimal solution. In this work, we propose a hybrid optimization approach that integrates Gated Recurrent Units (GRU), Convolutional Neural Networks (CNN), and a bilinear strategy as an innovative alternative to conventional optimizers for predicting optimal parameters of QAOA circuits. GRU serves to stochastically initialize favorable parameters for depth-1 circuits, while CNN predicts initial parameters for depth-2 circuits based on the optimized parameters of depth-1 circuits. To assess the efficacy of our approach, we conducted a comparative analysis with traditional initialization methods using QAOA on Erdős-Rényi graph instances, revealing superior optimal approximation ratios. We employ the bilinear strategy to initialize QAOA circuit parameters at greater depths, with reference parameters obtained from GRU-CNN optimization. This approach allows us to forecast parameters for a depth-12 QAOA circuit, yielding a remarkable approximation ratio of 0.998 across 10 qubits, which surpasses that of the random initialization strategy and the PPN2 method at a depth of 10. The proposed hybrid GRU-CNN bilinear optimization method significantly improves the effectiveness and accuracy of parameters initialization, offering a promising iterative framework for QAOA that elevates its performance.

On Fractional Discrete Financial System: Bifurcation, Chaos and Control

Abstract The dynamic analysis of systems financial is a developed field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional fractional financial map, we dissection of its nonlinear dynamics system under commensurate and incommensurate orders. Such, we evaluate when the equilibrium points are stable or unstable at various fractional order further we use many numerical methods, phase plots in 2D and 3D projections, bifurcation diagram and maximum Lyapunov exponent. These techniques reveal that the financial map exhibits chaotic attractor. This study grounded on the Caputo-like discrete operator, specifically influenced by the variance of the commensurate and incommensurate orders. Furthermore, we confirm the presence and measure complexity of chaos in the financial map by 0-1 test and the approximate entropy algorithm. Additionally, we offer non-linear type controllers so that to stabilize the fractional financial map. Finally, the numerical results of this study was done using MATLAB.

Scheduling jobs with shared additional operations on parallel identical machines

The paper is concerned with assigning jobs, each with associated additional operations, to identical machines. A machine, allocated to a job, must also process all the additional operations associated with this job. An additional operation that is associated with several jobs assigned to the same machine needs to be processed by this machine only once. The goal is to minimise the time needed for the completion of all jobs and their additional operations. It is shown that even very restricted particular cases of the considered problem remain NP-hard in the strong sense. For the general case, the paper introduces two mixed integer linear programs as well as a broad class of approximation algorithms and a performance guarantee that is valid for any algorithm in this class. It is shown that, for one of the above-mentioned NP-hard particular cases, the considered class contains the best possible approximation algorithm. The performance of the mixed integer linear programs and several approximation algorithms is compared by means of computational experiments.

A Comparative Study of Cat Swarm Algorithm for Graph Coloring Problem: Convergence Analysis and Performance Evaluation

The Graph Coloring Problem (GCP) is a significant optimization challenge widely suitable to solve scheduling problems. Its goal is to specify the minimum colors (k) required to color a graph properly. Due to its NP-completeness, exact algorithms become impractical for graphs exceeding 100 vertices. As a result, approximation algorithms have gained prominence for tackling large-scale instances. In this context, the Cat Swarm algorithm, a novel population-based metaheuristic in the domain of swarm intelligence, has demonstrated promising convergence properties compared to other population-based algorithms. This research focuses on designing and implementing the Cat Swarm algorithm to address the GCP. By conducting a comparative study with established algorithms, our investigation revolves around quantifying the minimum value of k required by the Cat Swarm algorithm for each graph instance. The evaluation metrics include the algorithm's running time in seconds, success rate, and the mean count of iterations or assessments required to reach a goal.

First-order reasoning and efficient semi-algebraic proofs

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

A Comparative Study of Cat Swarm Algorithm for Graph Coloring Problem: Convergence Analysis and Performance Evaluation

The Graph Coloring Problem (GCP) is a significant optimization challenge widely suitable to solve scheduling problems. Its goal is to specify the minimum colors (k) required to color a graph properly. Due to its NP-completeness, exact algorithms become impractical for graphs exceeding 100 vertices. As a result, approximation algorithms have gained prominence for tackling large-scale instances. In this context, the Cat Swarm algorithm, a novel population-based metaheuristic in the domain of swarm intelligence, has demonstrated promising convergence properties compared to other population-based algorithms. This research focuses on designing and implementing the Cat Swarm algorithm to address the GCP. By conducting a comparative study with established algorithms, our investigation revolves around quantifying the minimum value of k required by the Cat Swarm algorithm for each graph instance. The evaluation metrics include the algorithm's running time in seconds, success rate, and the mean count of iterations or assessments required to reach a goal.

Near-term quantum algorithm for solving the MaxCut problem with fewer quantum resources

MaxCut is an NP-hard combinatorial optimization problem in graph theory. The quantum approximate optimization algorithms (QAOAs) offer new methods for solving such problems, which are potentially better than classical schemes. However, the requirement of N qubits for solving graphs with N-vertex in QAOA, coupled with the large-N trainability issue due to barren plateaus, poses a substantial challenge for noisy intermediate-scale quantum (NISQ) devices. Recently, a hybrid quantum–classical algorithm has been proposed for solving semidefinite programs (SDPs), named NISQ SDP Solver (NSS). In this paper, we study the performance of the NSS for solving MaxCut problem via the introduction of a near-term quantum algorithm and execution of experimental simulations. Since the MaxCut problem admits a relaxation into an SDP formulation, the SDP can be solved using NSS, with a subsequent classical post-processing step converting the hybrid density matrix into a MaxCut solution. After performing these steps, the near-term quantum algorithm will also inherit the advantages of NSS. We analyze the algorithm as compared to QAOAs and find that the depth of the quantum circuit is independent of the number of edges on the graph. Our algorithm requires logN qubits and has O(1) circuit depth. This implies that it uses exponentially fewer qubits compared to QAOAs while also requiring a significantly reduced circuit depth to solve the MaxCut problem. To demonstrate the effectiveness of the NSS, we focused on 3-regular graphs, 9-regular graphs, and ER graphs. Numerical results indicate that the quantum algorithm achieves a comparable approximation ratio with classical methods by solving the original high dimensional problem within a lower dimensional ansatz space. Analysis under various initial states shows that the algorithm exhibits a remarkably stable performance.

Uniformly monotone partitioning of polygons

We study the problem of partitioning a simple polygon P with n vertices (including R reflex vertices) and no holes into a minimum number of uniformly monotone subpolygons using diagonals that lie inside P, each connecting two distinct vertices of P. A set of polygons is uniformly monotone if there exists a line such that all of the polygons are monotone with respect to the line. We present an O(nRlog⁡n+R5)-time algorithm for the problem. When Steiner points can be placed on the boundary of P and each Steiner point is considered as a vertex of P, we present an O(n+R5)-time algorithm for the problem. We present an O(n+R4)-time algorithm when Steiner points can be placed anywhere in P. Our algorithms improve upon the previously best ones. We also present simple and efficient 2-approximation algorithms.

Approximation algorithms for two clustered arc routing problems

Approximation algorithms for two clustered arc routing problems

Toward reliability-satisfied and hop-constrained backup paths for industrial internet demands

Industrial data transfer must rapidly recover from possible link failures to ensure reliability. The backup path mechanism is one of the transfer protections against multiple link failures. In this paper, we study a reliability-satisfied and hop-constrained backup path model that strives to reduce network capacity consumption under constraints on the reliability and maximum hop number of the backup path. The constraint on the maximum hop number is used to ensure the reliability and delay of a backup path. We propose an approximation algorithm to solve the dynamic backup path arrangement problem that is NP-hard.

A two-stage budget-feasible mechanism for mobile crowdsensing based on maximum user revenue routing

Through user participation, mobile crowdsensing (MCS) services overcome the problem of the excessive costs of relying solely on the active deployment of sensors and of achieving large-scale and low-cost applications of the Internet of Things, which is a research hotspot. However, current research on MCS issues adopts the perspective of service providers and does not consider user strategies, so the corresponding models cannot accurately reflect the complete status of the system. Therefore, this paper decomposes the MCS problem into a two-stage game process. By doing so, the strategies of both users and service providers can be considered, thus maximizing the interest for both parties. In the first stage, users determine the optimal route based on information released by the service provider. In the second stage, the service provider determines the winning users and the corresponding payment plan based on the route and bid information submitted by all users. Specifically, we express the user’s optimal route decision-making problem as a traveling salesman problem with time windows and node number constraints. Accordingly, we design the F-MAX-RR algorithm based on an evolutionary algorithm. We show that this algorithm can achieve an approximation ratio of (1−1/e), with the expected number of iterations being 8eL2(L+1)M. In the second stage, to maximize the total utility of the system, we transform the problem into an integer programming model with a budget constraint, which satisfies submodular characteristics. We design the S-MAX-TUM mechanism based on monotonic allocation and critical price theory to solve the problem of winning user decision-making and pricing. We demonstrate the economic characteristics of the mechanism, including truthfulness, individual rationality, and budget feasibility. The experimental results indicate the effective performance of the designed mechanisms.