Malleable Scheduling Beyond Identical Machines

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds

The energy scheduling problem: Industrial case-study and constraint propagation techniques

This paper deals with production scheduling involving energy constraints, typically electrical energy. We start by an industrial case-study for which we propose a two-step integer/constraint programming method. From the industrial problem we derive a generic problem,the Energy Scheduling Problem (EnSP). We propose an extension of specific resource constraint propagation techniques to efficiently prune the search space for EnSP solving. We also present a branching scheme to solve the problem via tree search.Finally,computational results are provided

Machine Scheduling with Resource Dependent Processing Times

We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;

Performance optimization and energy efficiency of big-data computing workflows

Next-generation e-science is producing colossal amounts of data, now frequently termed as Big Data, on the order of terabyte at present and petabyte or even exabyte in the predictable future. These scientific applications typically feature data-intensive workflows comprised of moldable parallel computing jobs, such as MapReduce, with intricate inter-job dependencies. The granularity of task partitioning in each moldable job of such big data workflows has a significant impact on workflow completion time, energy consumption, and financial cost if executed in clouds, which remains largely unexplored. This dissertation conducts an in-depth investigation into the properties of moldable jobs and provides an experiment-based validation of the performance model where the total workload of a moldable job increases along with the degree of parallelism. Furthermore, this dissertation conducts rigorous research on workflow execution dynamics in resource sharing environments and explores the interactions between workflow mapping and task scheduling on various computing platforms. A workflow optimization architecture is developed to seamlessly integrate three interrelated technical components, i.e., resource allocation, job mapping, and task scheduling. Cloud computing provides a cost-effective computing platform for big data workflows where moldable parallel computing models are widely applied to meet stringent performance requirements. Based on the moldable parallel computing performance model, a big-data workflow mapping model is constructed and a workflow mapping problem is formulated to minimize workflow makespan under a budget constraint in public clouds. This dissertation shows this problem to be strongly NP-complete and designs i) a fully polynomial-time approximation scheme for a special case with a pipeline-structured workflow executed on virtual machines of a single class, and ii) a heuristic for a generalized problem with an arbitrary directed acyclic graph-structured workflow executed on virtual machines of multiple classes. The performance superiority of the proposed solution is illustrated by extensive simulation-based results in Hadoop/YARN in comparison with existing workflow mapping models and algorithms. Considering that large-scale workflows for big data analytics have become a main consumer of energy in data centers, this dissertation also delves into the problem of static workflow mapping to minimize the dynamic energy consumption of a workflow request under a deadline constraint in Hadoop clusters, which is shown to be strongly NP-hard. A fully polynomial-time approximation scheme is designed for a special case with a pipeline-structured workflow on a homogeneous cluster and a heuristic is designed for the generalized problem with an arbitrary directed acyclic graph-structured workflow on a heterogeneous cluster. This problem is further extended to a dynamic version with deadline-constrained MapReduce workflows to minimize dynamic energy consumption in Hadoop clusters. This dissertation proposes a semi-dynamic online scheduling algorithm based on adaptive task partitioning to reduce dynamic energy consumption while meeting performance requirements from a global perspective, and also develops corresponding system modules for algorithm implementation in the Hadoop ecosystem. The performance superiority of the proposed solutions in terms of dynamic energy saving and deadline missing rate is illustrated by extensive simulation results in comparison with existing algorithms, and further validated through real-life workflow implementation and experiments using the Oozie workflow engine in Hadoop/YARN systems

Closing the Gap for Pseudo-Polynomial Strip Packing

Two-dimensional packing problems are a fundamental class of optimization problems and Strip Packing is one of the most natural and famous among them. Indeed it can be defined in just one sentence: Given a set of rectangular axis parallel items and a strip with bounded width and infinite height, the objective is to find a packing of the items into the strip minimizing the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial time algorithm for Strip Packing with a ratio better than 5/4 unless P = NP. The best algorithm so far has a ratio of 4/3 + epsilon. In this paper, we close the gap between inapproximability result and currently known algorithms by presenting an algorithm with approximation ratio 5/4 + epsilon. The algorithm relies on a new structural result which is the main accomplishment of this paper. It states that each optimal solution can be transformed with bounded loss in the objective such that it has one of a polynomial number of different forms thus making the problem tractable by standard techniques, i.e., dynamic programming. To show the conceptual strength of the approach, we extend our result to other problems as well, e.g., Strip Packing with 90 degree rotations and Contiguous Moldable Task Scheduling, and present algorithms with approximation ratio 5/4 + epsilon for these problems as well