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Integrated scheduling on a batch machine to minimize production, inventory and distribution costs

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2017年第1期258卷 104-112页

作者： Cheng, Ba-Yi Leung, Joseph Y-T. Li, Kai Hefei Univ Technol Sch Management Hefei 230009 Peoples R China Minist Educ Key Lab Proc Optimizat & Intelligent Decis Making Hefei 230009 Peoples R China New Jersey Inst Technol Dept Comp Sci Newark NJ 07012 USA

We consider the problem of scheduling a set of jobs on a single batch-processing machine. Each job has a size and a processing time. The jobs are batched together and scheduled on the batch-processing machine, provided that the total size does not exceed the machine capacity. The processing time of the batch is the longest processing time among all the jobs in the batch. There is a single vehicle to deliver the final products to the customer. If the vehicle has not returned, completed batches will be put into the inventory. In this paper, we consider the problem of minimizing the production, delivery and inventory costs. We show that if the jobs have the same size, there is an O(nlogn)-time algorithm to find an optimal solution. If the jobs have the same processing time, there is a fast approximation algorithm with an absolute worst-case ratio less than 1.783 and an asymptotic worst-case ratio equal to 11/9. When the jobs have arbitrary sizes and arbitrary processing times, there is a fast approximation algorithm with absolute and asymptotic worst-case ratios less than or equal to 2, respectively. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Batch-processing machines Production Inventory Distribution approximation algorithms

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ESTIMATING THE WEIGHT OF METRIC MINIMUM SPANNING TREES IN SUBLINEAR TIME

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SIAM JOURNAL ON COMPUTING 2009年第3期39卷 904-922页

作者： Czumaj, Artur Sohler, Christian Univ Warwick Dept Comp Sci Coventry CV4 7AL W Midlands England Univ Warwick Ctr Discrete Math & Applicat Coventry CV4 7AL W Midlands England Tech Univ Dortmund Dept Comp Sci D-44221 Dortmund Germany

In this paper we present a sublinear-time (1 + epsilon)-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an n-point metric space. The running time of the algorithm is (O) over tilde (n/epsilon(O(1))). Since the full description of an n-point metric space is of size Theta(n(2)), the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. We also show that no deterministic algorithm can achieve a B-approximation in o(n(2)/B-3) time. Furthermore, it has been previously shown that no o(n(2)) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.

关键词： randomized algorithms approximation algorithms sublinear-time algorithms

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Statistical Beamforming for FDD Downlink Massive MIMO via Spatial Information Extraction and Beam Selection

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2020年第7期19卷 4617-4631页

作者： Liu, Hang Yuan, Xiaojun Zhang, Ying Jun Chinese Univ Hong Kong Dept Informat Engn Hong Kong Peoples R China Univ Elect Sci & Technol China Ctr Intelligent Networking & Commun Chengdu 611731 Peoples R China

In this paper, we study the beamforming design problem in frequency-division duplexing (FDD) downlink massive MIMO systems, where instantaneous channel state information (CSI) is assumed to be unavailable at the base station (BS). We propose to extract the information of the angle-of-departures (AoDs) and the corresponding large-scale fading coefficients (a.k.a. spatial information) of the downlink channel from the uplink channel estimation procedure, based on which a novel downlink beamforming design is presented. By separating the subpaths for different users based on the spatial information and the hidden sparsity of the physical channel, we construct near-orthogonal virtual channels in the beamforming design. Furthermore, we derive a sum-rate expression and its approximations for the proposed system. Based on these closed-form rate expressions, we develop two low-complexity beam selection schemes and carry out asymptotic analysis to provide valuable insights on the system design. Numerical results demonstrate a significant performance improvement of our proposed algorithm over the state-of-the-art beamforming approach.

关键词： Downlink Array signal processing Uplink MIMO communication Training Channel estimation approximation algorithms Massive MIMO FDD statistical beamforming spatial reciprocity

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Cooperation in multi-organization scheduling

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CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE 2009年第7期21卷 905-921页

作者： Pascual, Fanny Rzadca, Krzysztof Trystram, Denis Grenoble Univ LIG F-38330 Montbonnot St Martin France Univ Paris 06 LIP6 F-75016 Paris France Polish Japanese Inst Informat Technol PL-02008 Warsaw Poland

The distributed nature of the grid results in the problem of scheduling parallel jobs produced by several independent organizations that have partial control over the system. We consider systems in which each organization owns a cluster of processors. Each organization wants its tasks to be completed as soon as possible. In this paper, we model an off-line system consisting of N identical clusters of m processors. We show that it is always possible to produce a collaborative solution that respects participants' selfish goals, at the same time improving the global performance of the system. We propose an algorithm (called MOLBA) with a guaranteed worst-case performance ratio on the global makespan, equal to 4. Next, we show that a better bound (equal to 3) can be obtained in a specific case when the last completed job requires at most m/2 processors. Then, we derive another algorithm (called ILBA) that in practice improves the proposed, guaranteed solution by further balancing the schedules. Finally, by an extensive evaluation bit simulation, we show that the algorithms are efficient on typical instances. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： grid scheduling approximation algorithms multi-objective optimization

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An Efficient Approximate Algorithm for Disjoint QoS Routing

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MATHEMATICAL PROBLEMS IN ENGINEERING 2013年第1期2013卷 1-9页

作者： Yu, Zhanke Ma, Feng Liu, Jingxia Hu, Bingxin Zhang, Zhaodong PLA Univ Sci & Technol Coll Commun Engn Nanjing 210007 Jiangsu Peoples R China

Disjoint routing is used to find the disjoint paths between a source and a destination subject to QoS requirements. Disjoint QoS routing is an effective strategy to achieve robustness, load balancing, congestion reduction, and an increased throughput in computer networks. For multiple additive constraints, disjoint QoS routing is an NP-complete class that cannot be exactly solved in polynomial time. In the paper, the disjoint QoS routing problem was formulated as a 0-1 integer linear programming. The complicating constraints were included in the objective function using an adaptive penalty function. The special model with a totally unimodular constraint coefficient matrix was constructed and could be solved rapidly as a linear programming. An efficient algorithm using an adaptive penalty function and 0-1 integer linear programming for the disjoint QoS routing problems was designed. The proposed algorithm could obtain the optimal solution, considerably reducing the computational time consumption and improving the computational efficiency. Theoretical analysis and simulation experiments were performed to evaluate the proposed algorithm performance. Through the establishment of random network topologies using Matlab, the average running time, the optimal objective value, and the success rate were evaluated based on the optimal values obtained in Cplex. The simulation experiments validated the effectiveness of the proposed heuristic algorithm.

关键词： approximation algorithms ROUTING (Computer network management) QUALITY of service ROBUST control LOAD balancing (Computer networks) COMPUTER networks LINEAR programming

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BERN-NN-IBF: Enhancing Neural Network Bound Propagation Through Implicit Bernstein Form and Optimized Tensor Operations

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2024年第11期43卷 4334-4345页

作者： Fatnassi, Wael Feeney, Arthur Yamamoto, Valen Chandramowlishwaran, Aparna Shoukry, Yasser Univ Calif Irvine Dept Elect Engn & Comp Sci Irvine CA 92697 USA

Neural networks have emerged as powerful tools across various domains, exhibiting remarkable empirical performance that motivated their widespread adoption in safety-critical applications, which, in turn, necessitates rigorous formal verification techniques to ensure their reliability and robustness. Tight bound propagation plays a crucial role in the formal verification process by providing precise bounds that can be used to formulate and verify properties, such as safety, robustness, and fairness. While state-of-the-art tools use linear and convex approximations to compute upper/lower bounds for each neuron's outputs, recent advances have shown that nonlinear approximations based on Bernstein polynomials lead to tighter bounds but suffer from scalability issues. To that end, this article introduces BERN-NN-IBF, a significant enhancement of the Bernstein-polynomial-based bound propagation algorithms. BERN-NN-IBF offers three main contributions: 1) a memory-efficient encoding of Bernstein polynomials to scale the bound propagation algorithms;2) optimized tensor operations for the new polynomial encoding to maintain the integrity of the bounds while enhancing computational efficiency;and 3) tighter under-approximations of the ReLU activation function using quadratic polynomials tailored to minimize approximation errors. Through comprehensive testing, we demonstrate that BERN-NN-IBF achieves tighter bounds and higher computational efficiency compared to the original BERN-NN and state-of-the-art methods, including linear and convex programming used within the winner of the VNN-COMPETITION.

关键词： Tensors Memory management approximation algorithms Polynomials Robustness Encoding Computational efficiency Biological neural networks Formal verification Testing model checking neural networks

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SIMULTANEOUS POLE ASSIGNMENT IN LINEAR PERIODIC-SYSTEMS BY CONSTRAINED STRUCTURE FEEDBACK

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1989年第2期34卷 168-173页

作者： JUAN, YC KABAMBA, PT Department of Aerospace Engineering University of Michigan Ann Arbor MI USA

The authors present methods for pole assignment by feedback with constrained structure in linear periodic systems using generalized sampled-data hold functions (GSHF). Constrained structure is taken to mean that the input of each channel is restricted to depend only on the measurements of some specific output channels. The basic idea of GSHF control is to use sample and hold, but to consider the hold function as a design parameter. Four strategies are proposed for closing a control loop using GSHF. The key to the results obtained is that, when GSHF control with constrained structure is applied to a periodic system, a discrete-time time-invariant, decentralized system is obtained for which control design methods are available.< >

关键词： Periodic structures Feedback Iterative algorithms Eigenvalues and eigenfunctions Robustness Optimal control approximation algorithms Transfer functions Programming Frequency domain analysis

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A Novel Real-Time Filtering Method to General Nonlinear Filtering Problem Without Memory

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IEEE ACCESS 2021年 9卷 119343-119352页

作者： Shi, Ji Chen, Xiuqiong Yau, Stephen Shing-Toung Capital Normal Univ Beijing Adv Innovat Ctr Imaging Theory & Technol Acad Multidisciplinary Studies Beijing 100048 Peoples R China Tsinghua Univ Yau Math Sci Ctr Beijing 100084 Peoples R China Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China Yanqi Lake Beijing Inst Math Sci & Applicat Beijing 101408 Peoples R China

In this paper, the filtering problem for the general time-invariant nonlinear state-observation system is considered. Our work is based on the Yau-Yau filtering framework developed by S.-T. Yau and the third author in 2008. The key problem of Yau-Yau filtering framework is how to compute the solution to forward Kolmogorov equation (FKE) off-line effectively. Motivated by the supervised learning in machine learning, we develop an efficient method to numerically solve the FKE off-line from the point of view of optimization. Specifically, for the off-line computation part, the computation of the solution to a FKE is reduced to computing a linear system of equations by making the temporal inverse transformation and the loss function optimization, and we store the results for the preparation of on-line computation. For the on-line computation part, the unnormalized density function is approximated by a complete polynomial basis, and then the estimation of the state is computed using the stored off-line data. Our method has the merits of easily implementing, real-time and memoryless. More importantly, it can be applicable for moderate-high dimensional cases. Numerical experiments have been carried out to verify the feasibility of our method. Our algorithm outperforms extended Kalman filter, unscented Kalman filter and particle filter both in accuracy and costing time.

关键词： Real-time systems Kalman filters approximation algorithms Mathematical model Density functional theory Supervised learning Optimization Duncan-Mortensen-Zakai equation nonlinear filtering forward Kolmogorov equation estimation theory numerical algorithms

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The General Steiner Tree-Star problem

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INFORMATION PROCESSING LETTERS 2002年第4期84卷 215-220页

作者： Khuller, S Zhu, A Univ Maryland Dept Comp Sci College Pk MD 20742 USA Univ Maryland Inst Adv Comp Studies College Pk MD 20742 USA Stanford Univ Dept Comp Sci Stanford CA 94305 USA

The Steiner tree problem is defined as follows-given a graph G = (V, E) and a subset X C V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a "switch" cost in addition to the cost of the edges. Switches are placed at internal nodes of the tree (essentially, we may assume that all non-leaf nodes of the Steiner tree have a switch). The cost for placing a switch may vary from node to node. A restricted version of this problem, where the terminal set X cannot be connected to each other directly but only via the Steiner nodes V \ X, is referred to as the Steiner Tree-Star problem. The General Steiner Tree-Star problem does not require the terminal set and Steiner node set to be disjoint. This generalized problem can be reduced to the node weighted Steiner tree problem, for which algorithms with performance guarantees of 0 (Inn) are known. However, such approach does not make use of the fact that the edge costs form a metric. In this paper we derive approximation algorithms with small constant factors for this problem. We show two different polynomial time algorithms with approximation factors of 5.16 and 5. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： approximation algorithms facility location Steiner trees

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The UGC Hardness Threshold of the L_p Grothendieck Problem

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MATHEMATICS OF OPERATIONS RESEARCH 2010年第2期35卷 267-283页

作者： Kindler, Guy Naor, Assaf Schechtman, Gideon Hebrew Univ Jerusalem Sch Comp Sci & Engn IL-91904 Jerusalem Israel NYU Courant Inst New York NY 10012 USA Weizmann Inst Sci Dept Math IL-76100 Rehovot Israel

For p >= 2 we consider the problem of, given an n x n matrix A = (a(ij)) whose diagonal entries vanish, approximating in polynomial time the number Opt(p)(A):= max{(i,j=1)Sigma(n) a(ij)x(i)x(j): (x(1), ..., x(n)) is an element of R-n boolean AND ((i=1)Sigma(n) vertical bar x(i)vertical bar(p))(1/p) <= 1}. When p = 2 this is simply the problem of computing the maximum eigenvalue of A, whereas for p = infinity ( actually it suffices to take p approximate to log n) it is the Grothendieck problem on the complete graph, which was shown to have a O(log n) approximation algorithm in Nemirovski et al. [Nemirovski, A., C. Roos, T. Terlaky. 1999. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. Ser. A 86(3) 463-473], Megretski [Megretski, A. 2001. Relaxations of quadratic programs in operator theory and system analysis. Systems, approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Vol. 129. Operator Theory Advances and Applications. Birkhauser, Basel, 365-392], Charikar and Wirth [Charikar, M., A. Wirth. 2004. Maximizing quadratic programs: Extending Grothendieck's inequality. Proc. 45th Annual Sympos. Foundations Comput. Sci., IEEE Computer Society, 54-60] and was used in the work of Charikar and Wirth noted above, to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering. Thus the problem of approximating Opt(p)(A) interpolates between the spectral ( p = 2) case and the correlation clustering (p=infinity) case. From a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard-wall potential well. We design a polynomial time algorithm which, given p >= 2 and an n x n matrix A = (a(ij)) with zeros on the diagonal, computes Opt(p)(A) up to a factor p/e + 30 log p. On the other hand, assuming the unique games conjecture ( UGC) we show that it is NP-hard to approximate Opt(p)(A) up to a factor smaller

关键词： quadratic programming approximation algorithms unique games hardness

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