39 research outputs found

    Speed-scaling with no Preemptions

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    We revisit the non-preemptive speed-scaling problem, in which a set of jobs have to be executed on a single or a set of parallel speed-scalable processor(s) between their release dates and deadlines so that the energy consumption to be minimized. We adopt the speed-scaling mechanism first introduced in [Yao et al., FOCS 1995] according to which the power dissipated is a convex function of the processor's speed. Intuitively, the higher is the speed of a processor, the higher is the energy consumption. For the single-processor case, we improve the best known approximation algorithm by providing a (1+Ï”)αB~α(1+\epsilon)^{\alpha}\tilde{B}_{\alpha}-approximation algorithm, where B~α\tilde{B}_{\alpha} is a generalization of the Bell number. For the multiprocessor case, we present an approximation algorithm of ratio B~α((1+Ï”)(1+wmax⁥wmin⁥))α\tilde{B}_{\alpha}((1+\epsilon)(1+\frac{w_{\max}}{w_{\min}}))^{\alpha} improving the best known result by a factor of (52)α−1(wmax⁥wmin⁥)α(\frac{5}{2})^{\alpha-1}(\frac{w_{\max}}{w_{\min}})^{\alpha}. Notice that our result holds for the fully heterogeneous environment while the previous known result holds only in the more restricted case of parallel processors with identical power functions

    Heuristics with Performance Guarantees for the Minimum Number of Matches Problem in Heat Recovery Network Design

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    Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an NP-hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods

    Politiques de gestion d’Énergie et de TempĂ©rature dans les SystĂšmes Informatiques

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    Nowadays, the energy consumption and the heat dissipation of computing environmentshave emerged as crucial issues. Indeed, large data centers consume as much electricityas a city while modern processors attain high temperatures degrading their performanceand decreasing their reliability. In this thesis, we study various energy and temperatureaware scheduling problems and we focus on their complexity and approximability.A dominant technique for saving energy is by proper scheduling of the jobs through theoperating system combined with appropriate scaling of the processor’s speed. This techniqueis referred to as speed scaling in the literature. The theoretical study of speed scalingwas initiated by Yao, Demers and Shenker (1995) who considered the single-processorproblem of scheduling preemptively a set of jobs, each one specified by an amount ofwork, a release date and a deadline, so as to minimize the total energy consumption.In order to measure the energy consumption of a processor, the authors considered thewell-known rule according to which the processor’s power consumption is P(t) = s(t)α ateach time t, where s(t) is the processor’s speed at t and α > 1 is a machine-dependentconstant (usually α ∈ [2, 3]). Here, we study speed scaling problems on a single processor,on homogeneous parallel processors, on heterogeneous environments and on shopenvironments. In most cases, the objective is the minimization of the energy but we alsoaddress problems in which we are interested in capturing the trade-off between energyand performance.We tackle speed scaling problems through different approaches. For non-preemptiveproblems, we explore the idea of transforming optimal preemptive schedules to nonpreemptiveones. Moreover, we exploit the fact that some problems can be formulatedas convex programs and we propose greedy algorithms that produce optimal solutionssatisfying the KKT conditions which are necessary and sufficient for optimality in convexprogramming. In the context of convex programming and KKT conditions, we also studythe design of primal-dual algorithms. Additionally, we solve speed scaling problems byformulating them as convex cost flow or minimum weighted bipartite matching problems.Finally, we elaborate on approximating energy minimization problems that can be formulatedas integer configuration linear programs. We can obtain an approximate solutionfor such a problem by solving the fractional relaxation of an integer configuration linearprogram for it and applying randomized rounding.In this thesis, we solve some new energy aware scheduling problems and we improvethe best-known algorithms for some other problems. For instance, we improve the bestknownapproximation algorithm for the single-processor non-preemptive energy minimizationproblem which is strongly NP-hard. When α = 3, we decrease the approximationratio from 2048 to 20. Furthermore, we propose a faster optimal combinatorial algorithmviiviiifor the preemptive migratory energy minimization problem on power-homogeneous processors,while the best-known algorithm was based on solving linear programs. Last butnot least, we improve the best-known approximation algorithm for the preemptive nonmigratoryenergy minimization problem on power-homogeneous processors for fractionalvalues of α. Our algorithm can be applied even in the more general case where the processorsare heterogeneous and, for αmax = 2.5 (which is the maximum constant α amongall processors), we get an improvement of the approximation ratio from 5 to 3.08.In order to manage the thermal behavior of a computing device, we adopt the approachof Chrobak, DĂŒrr, Hurand and Robert (2011). The main assumption is that some jobsare more CPU intensive than others and more heat is generated during their execution.So, each job is associated with a heat contribution which is the impact of the job on theprocessor’s temperature. In this setting, we study the complexity and the approximabilityof multiprocessor scheduling problems where either there is a constraint on the processors’temperature and our aim is to optimize some performance metric or the temperature isthe optimization goal itself.La gestion de la consommation d’énergie et de la tempĂ©rature est devenue un enjeucrucial dans les systĂšmes informatiques. En effet, un grand centre de donnĂ©es consommeautant d’électricitĂ© qu’une ville et les processeurs modernes atteignent des tempĂ©raturesimportantes dĂ©gradant ainsi leurs performances et leur fiabilitĂ©. Dans cette thĂšse, nousĂ©tudions diffĂ©rents problĂšmes d’ordonnancement prenant en compte la consommationd’énergie et la tempĂ©rature des processeurs en se focalisant sur leur complexitĂ© et leurapproximabilitĂ©. Pour cela, nous utilisons le modĂšle de Yao et al. (1995) (modĂšle devariation de vitesse) pour la gestion d’énergie et le modĂšle de Chrobak et al. (2008) pourla gestion de la tempĂ©rature

    Energy Efficient Scheduling and Routing via Randomized Rounding

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    We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same quality as for the homogeneous environment. By exploiting the result for the preemptive-non-migratory variant, we are able to improve the best known approximation ratio for the single processor non-preemptive problem. Furthermore, we show that our approach allows to obtain a constant-factor approximation algorithm for the power-aware preemptive job shop scheduling problem. Finally, we consider the min-power routing problem where we are given a network modeled by an undirected graph and a set of uniform demands that have to be routed on integral routes from their sources to their destinations so that the energy consumption is minimized. We improve the best known approximation ratio for this problem.Comment: 27 page

    Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

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    Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance

    Energy Efficient Scheduling of MapReduce Jobs

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    MapReduce is emerged as a prominent programming model for data-intensive computation. In this work, we study power-aware MapReduce scheduling in the speed scaling setting first introduced by Yao et al. [FOCS 1995]. We focus on the minimization of the total weighted completion time of a set of MapReduce jobs under a given budget of energy. Using a linear programming relaxation of our problem, we derive a polynomial time constant-factor approximation algorithm. We also propose a convex programming formulation that we combine with standard list scheduling policies, and we evaluate their performance using simulations.Comment: 22 page

    Optimal energy trade-off schedules

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    International audienceWe consider scheduling tasks that arrive over time on a speed scalable processor. At each time a schedule specifies a job to be run and the speed at which the processor is run. Processors are generally less energy efficient at higher speeds. We seek to understand the structure of schedules that optimally trade-off the energy used by the processor with a common scheduling quality of service measure, fractional weighted delay. We assume that there is some user defined parameter ÎČ specifying the user's desired additive trade-off between energy efficiency and quality of service. We prove that the optimal energy trade-off schedule is essentially unique, and has a simple structure. Thus it is easy to check the optimality of a schedule. We further prove that the optimal energy trade-off schedule changes continuously as a function of the parameter ÎČ. Thus it is possible to compute the optimal energy trade-off schedule using a natural homotopic optimization algorithm. We further show that multiplicative trade-off schedules have fewer desirable properties
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