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Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the minimax approximation on a prescribed linear functional space. Lawson’s iteration is a classical and well-known method for the task. However, Lawson’s iteration converges only linearly and in many cases, the convergence is very slow. In this paper, relying upon the Lagrange duality, we establish an L q -weighted dual programming for the discrete linear Chebyshev approximation. In this framework of dual problem, we revisit the convergence of Lawson’s iteration and provide a new and self-contained proof for the well-known Alternation Theorem in the real case; moreover, we propose a Newton type iteration, the interior-point method, to solve the L 2 -weighted dual programming. Numerical experiments are reported to demonstrate its fast convergence and its capability in finding the reference points that characterize the unique minimax approximation. [ABSTRACT FROM AUTHOR]

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a (2 + ε) -approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are δ -large, that is, have at least one edge whose length is at least δ , while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network. [ABSTRACT FROM AUTHOR]

This paper considers the basic problem of scheduling jobs online with preemption to maximize the number of jobs completed by their deadline on m identical machines. The main result is an O(1) competitive deterministic algorithm for any number of machines m > 1 . [ABSTRACT FROM AUTHOR]

Cloud data centers face various challenges, such as high energy consumption, environmental impact, and quality of service (QoS) requirements. Dynamic virtual machine (VM) consolidation is an effective approach to address these challenges, but it is a complex optimization problem that involves trade-offs between energy efficiency and QoS satisfaction. Moreover, the workload patterns in cloud data centers are often non-stationary and unpredictable, which makes it difficult to model them. In this paper, we propose a new method for dynamic VM consolidation that optimizes both energy efficiency and QoS objectives. Our approach is based on Markov chains and the artificial feeding birds (AFB) algorithm. Markov chains are used to model the resource utilization of each individual VM and PM based on the changes that happen in workload data. AFB algorithm is a metaheuristic optimization technique that mimics the behavior of birds in nature. We modify the AFB algorithm to suit the characteristics of the VM placement problem and to provide QoS-aware and energy-efficient solutions. Our approach also employs an online step detection method to capture variations in workload patterns. Furthermore, we introduce a new policy for VM selection from overloaded hosts, which considers the abrupt changes in the utilization processes of the VMs. The proposed algorithms are evaluated extensively using the CloudSim Toolkit with real workload data. The proposed system outperforms evaluation policies in multiple metrics, including energy consumption, SLA violations, and other essential metrics. [ABSTRACT FROM AUTHOR]

Kovaleva and Spieksma (SIAM J Discrete Math 20(3):48–768, 2006) considered the problem of stabbing a given set of horizontal line segments with the smallest number of horizontal and vertical lines. The standard LP relaxation for this problem is easily shown to have an integrality gap of at most 2 by treating the horizontal and vertical lines separately. However, Kovaleva and Spieksma observed that threshold rounding can be used to obtain an integrality gap of e / (e - 1) ≈ 1.58 which is also shown to be tight. This is one of the rare known examples where the obvious upper bound of 2 on the integrality gap of the standard LP relaxation can be improved. Our goal in this paper is to extend their proof to two other problems where the goal is to stab a set R of objects with horizontal and vertical lines: in the first problem R is a set of horizontal and vertical line segments, and in the second problem R is a set of unit sized squares. The proof of Kovaleva and Spieksma essentially shows the existence of an appropriate threshold which yields the improved approximation factor. We begin by showing that a random threshold picked from an appropriate distribution works. This reduces the problem to finding an appropriate distribution for a desired approximation ratio. In the first problem, we show that the required distribution can be found by solving a linear program. In the second problem, while it seems harder to find the optimal distribution, we show that using the uniform distribution an improved approximation factor can still be obtained by solving a number of linear programs. [ABSTRACT FROM AUTHOR]

Submodular maximization is a NP-hard combinatorial optimization problem regularly used in machine learning and data mining with large-scale data sets. To quantify the running time of approximation algorithms, the query complexity and adaptive complexity are two important measures, where the adaptivity of an algorithm is the number of sequential rounds it makes when each round can execute polynomially many function evaluations in parallel. These two concepts reasonably quantify the efficiency and practicability of parallel computation. In this paper, we consider the problem of maximizing a nonnegative monotone DR-submodular function over a bounded integer lattice with a cardinality constraint in a value oracle model. Prior to our work, Soma and Yoshida (Math Program 172:539–563, 2018) have studied this problem and present an approximation algorithm with almost optimal approximate ratio and the same adaptivity and query complexity. We develop two novel algorithms, called low query algorithm and low adaptivity algorithm, which have approximation ratios equal to Soma and Yoshida (2018), but reach a new record of query complexity and adaptivity. As for methods, compared with the threshold greedy in Soma and Yoshida (2018), the low query algorithm reduces the number of possible thresholds and improves both the adaptivity and query complexity. The low adaptivity algorithm further integrates a vector sequencing technique to accelerate adaptive complexity in an exponential manner while only making logarithmic sacrifices on oracle queries. [ABSTRACT FROM AUTHOR]

The numerical solution of boundary value problems by the finite element method leads to a system of equations with a matrix that depends on the triangulation mesh. A change in the geometric configuration of the mesh leads to the changes in the corresponding matrix and hence to the changes in the rate of convergence of the process of successive approximations to the numerical solution of the problem. The paper shows how it is possible to speed up the rate of convergence of the iterative solution process without changing the position of the grid vertices. A class of matrices is found whose mesh optimization leads to Delaunay triangulation. An example of using the results of the work in the numerical solution of the magnetic field is given. [ABSTRACT FROM AUTHOR]