Nützliche Strukturen und wie sie zu finden sind: Nicht Approximierbarkeit und Approximationen für diverse Varianten des Parallel Task Scheduling Problems

In this thesis, we consider the Parallel Task Scheduling problem and several variants. This problem and its variations have diverse applications in theory and practice; for example, they appear as sub-problems in higher dimensional problems. In the Parallel Task Scheduling problem, we are given a set of jobs and a set of identical machines. Each job is a parallel task; i.e., it needs a fixed number of identical machines to be processed. A schedule assigns to each job a set of machines it is processed on and a starting time. It is feasible if at each point in time each machine processes at most one job. In a variant of this problem, called Strip Packing, the identical machines are arranged in a total order, and jobs can only allocate neighboring machines with regard to this total order. In this case, we speak of Contiguous Parallel Task Scheduling as well. In another variant, called Single Resource Constraint Scheduling, we are given an additional constraint on how many jobs can be processed at the same time. For these variants of the Parallel Task Scheduling problem, we consider an extension, where the set of machines is grouped into identical clusters. When scheduling a job, we are allowed to allocate machines from only one cluster to process the job. For all these considered problems, we close some gaps between inapproximation or hardness result and the best possible algorithm. For Parallel Task Scheduling we prove that it is strongly NP-hard if we are given precisely 4 machines. Before it was known that it is strongly NP-hard if we are given at least 5 machines, and there was an (exact) pseudo-polynomial time algorithm for up to 3 machines. For Strip Packing, we present an algorithm with approximation ratio (5/4 +ε) and prove that there is no approximation with ratio less than 5/4 unless P = NP. Concerning Single Resource Constraint Scheduling, it is not possible to find an algorithm with ratio smaller than 3/2, unless P = NP, and we present an algorithm with ratio (3/2 +ε). For the extensions to identical clusters, there can be no approximation algorithm with a ratio smaller than 2 unless P = NP. For the extensions of Strip Packing and Parallel Task Scheduling there are 2-approximations already, but they have a huge worst case running time. We present 2-approximations that have a linear running time for the extensions of Strip Packing, Parallel Task Scheduling, and Single Resource Constraint Scheduling for the case that at least three clusters are present and greatly improve the running time for two clusters. Finally, we consider three variants of Scheduling on Identical Machines with setup times. We present EPTAS results for all of them which is the best one can hope for since these problems are strongly NP-complete.In dieser Thesis untersuchen wir das Problem Parallel Task Scheduling und einige seiner Varianten. Dieses Problem und seine Variationen haben vielfältige Anwendungen in Theorie und Praxis. Beispielsweise treten sie als Teilprobleme in höherdimensionalen Problemen auf. Im Problem Parallel Task Scheduling erhalten wir eine Menge von Jobs und eine Menge identischer Maschinen. Jeder Job ist ein paralleler Task, d. h. er benötigt eine feste Anzahl der identischen Maschinen, um bearbeitet zu werden. Ein Schedule ordnet den Jobs die Maschinen zu, auf denen sie bearbeitet werden sollen, sowie einen festen Startzeitpunkt der Bearbeitung. Der Schedule ist gültig, wenn zu jedem Zeitpunkt jede Maschine höchstens einen Job bearbeitet. Beim Strip Packing Problem sind die identischen Maschinen in einer totalen Ordnung angeordnet und Jobs können nur benachbarte Maschinen in Bezug auf diese Ordnung nutzen. In dem Single Resource Constraint Scheduling Problem gibt es eine zusätzliche Einschränkung, wie viele Jobs gleichzeitig verarbeitet werden können. Für die genannten Varianten des Parallel Task Scheduling Problems betrachten wir eine Erweiterung, bei der die Maschinen in identische Cluster gruppiert sind. Bei der Bearbeitung eines Jobs dürfen in diesem Modell nur Maschinen aus einem Cluster genutzt werden. Für all diese Probleme schließen wir Lücken zwischen Nichtapproximierbarkeit und Algorithmen. Für Parallel Task Scheduling zeigen wir, dass es stark NP-vollständig ist, wenn genau 4 Maschinen gegeben sind. Vorher war ein pseudopolynomieller Algorithmus für bis zu 3 Maschinen bekannt, sowie dass dieses Problem stark NP-vollständig ist für 5 oder mehr Maschinen. Für Strip Packing zeigen wir, dass es keinen pseudopolynomiellen Algorithmus gibt, der eine Güte besser als 5/4 besitzt und geben einen pseudopolynomiellen Algorithmus mit Güte (5/4 +ε) an. Für Single Resource Constraint Scheduling ist die bestmögliche Güte eine 3/2-Approximation und wir präsentieren eine (3/2 +ε)-Approximation. Für die Erweiterung auf identische Cluster gibt es keine Approximation mit Güte besser als 2. Vor unseren Untersuchungen waren bereits Algorithmen mit Güte 2 bekannt, die jedoch gigantische Worst-Case Laufzeiten haben. Wir geben für alle drei Varianten 2-Approximationen mit linearer Laufzeit an, sofern mindestens drei Cluster gegeben sind. Schlussendlich betrachten wir noch Scheduling auf Identischen Maschinen mit Setup Zeiten. Wir entwickeln für drei untersuche Varianten dieses Problems jeweils einen EPTAS, wobei ein EPTAS das beste ist, auf das man hoffen kann, es sei denn es gilt P = NP

Closing the Gap for Single Resource Constraint Scheduling

In the problem called single resource constraint scheduling, we are given m identical machines and a set of jobs, each needing one machine to be processed as well as a share of a limited renewable resource R. A schedule of these jobs is feasible if, at each point in the schedule, the number of machines and resources required by jobs processed at this time is not exceeded. It is NP-hard to approximate this problem with a ratio better than 3/2. On the other hand, the best algorithm so far has an absolute approximation ratio of 2+?. In this paper, we present an algorithm with absolute approximation ratio (3/2+?), which closes the gap between inapproximability and best algorithm with exception of a negligible small ?

Co-Scheduling Algorithms for High-Throughput Workload Execution

This paper investigates co-scheduling algorithms for processing a set of parallel applications. Instead of executing each application one by one, using a maximum degree of parallelism for each of them, we aim at scheduling several applications concurrently. We partition the original application set into a series of packs, which are executed one by one. A pack comprises several applications, each of them with an assigned number of processors, with the constraint that the total number of processors assigned within a pack does not exceed the maximum number of available processors. The objective is to determine a partition into packs, and an assignment of processors to applications, that minimize the sum of the execution times of the packs. We thoroughly study the complexity of this optimization problem, and propose several heuristics that exhibit very good performance on a variety of workloads, whose application execution times model profiles of parallel scientific codes. We show that co-scheduling leads to to faster workload completion time and to faster response times on average (hence increasing system throughput and saving energy), for significant benefits over traditional scheduling from both the user and system perspectives

Scheduling, Characterization and Prediction of HPC Workloads for Distributed Computing Environments

As High Performance Computing (HPC) has grown considerably and is expected to grow even more, effective resource management for distributed computing sys- tems is motivated more than ever. As the computational workloads grow in quantity, it is becoming more crucial to apply efficient resource management and workload scheduling to use resources efficiently while keeping the computational performance reasonably good. The problem of efficiently scheduling workloads on resources while meeting performance standards is hard. Additionally, non-clairvoyance of job dimen- sions makes resource management even harder in real-world scenarios. Our research methodology investigates the scheduling problem compliant for HPC and researches the challenges for deploying the scheduling in real world-scenarios using state of the art machine learning and data science techniques.To this end, this Ph.D. dissertation makes the following core contributions: a) We perform a theoretical analysis of space-sharing, non-preemptive scheduling: we studied this scheduling problem and proposed scheduling algorithms with polyno- mial computation time. We also proved constant upper-bounds for the performance of these algorithms. b) We studied the sensitivity of scheduling algorithms to the accuracy of runtime and devised a meta-learning approach to estimate prediction accuracy for newly submitted jobs to the HPC system. c) We studied the runtime prediction problem for HPC applications. For this purpose, we studied the distri- bution of available public workloads and proposed two different solutions that can predict multi-modal distributions: switching state-space models and Mixture Density Networks. d) We studied the effectiveness of recent recurrent neural network models for CPU usage trace prediction for individual VM traces as well as aggregate CPU usage traces. In this dissertation, we explore solutions to improve the performance of scheduling workloads on distributed systems.We begin by looking at the problem from the theoretical perspective. Modeling the problem mathematically, we first propose a scheduling algorithm that finds a constant approximation of the optimal solution for the problem in polynomial time. We prove that the performance of the algorithm (average completion time is the constant approximation of the performance of the optimal scheduling. We next look at the problem in real-world scenarios. Considering High-Performance Computing (HPC) workload computing environments as the most similar real-world equivalent of our mathematical model, we explore the problem of predicting application runtime. We propose an algorithm to handle the existing uncertainties in the real world and show-case our algorithm with demonstrative effectiveness in terms of response time and resource utilization. After looking at the uncertainty problem, we focus on trying to improve the accuracy of existing prediction approaches for HPC application runtime. We propose two solutions, one based on Kalman filters and one based on deep density mixture networks. We showcase the effectiveness of our prediction approaches by comparing with previous prediction approaches in terms of prediction accuracy and impact on improving scheduling performance. In the end, we focus on predicting resource usage for individual applications during their execution. We explore the application of recurrent neural networks for predicting resource usage of applications deployed on individual virtual machines. To validate our proposed models and solutions, we performed extensive trace-driven simulation and measured the effectiveness of our approaches

Optimizing and Reoptimizing: tackling static and dynamic combinatorial problems

As suggested by the title, in this thesis both static and dynamic problems of Operations Research will be addressed by either designing new procedures or adapting well-known algorithmic schemes. Specifically, the first part of the thesis is devoted to the discussion of three variants of the widely studied Shortest Path Problem, one of which is defined on dynamic graphs. Namely, first the Reoptimization of Shortest Paths in case of multiple and generic cost changes is dealt with an exact algorithm whose performance is compared with Dijkstra's label setting procedure in order to detect which approach has to be preferred. Secondly, the k-Color Shortest Path Problem is tackled. It is a recent problem, defined on an edge-constrained graph, for which a Dynamic Programming algorithm is proposed here; its performance is compared with the state of the art solution approach, namely a Branch & Bound procedure. Finally, the Resource Constrained Clustered Shortest Path Tree Problem is presented. It is a newly defined problem for which both a mathematical model and a Branch & Price procedure are detailed here. Moreover, the performance of this solution approach is compared with that of CPLEX solver. Furthermore, in the first part of the thesis, also the Path Planning in Urban Air Mobility, is discussed by considering both the definition of the Free-Space Maps and the computation of the trajectories. For the former purpose, three different but correlated discretization methods are described; as for the latter, a two steps resolution, offline and online, of the resulting shortest path problems is performed. In addition, it is checked whether the reoptimization algorithm can be used in the online step. In the second part of this thesis, the recently studied Additive Manufacturing Machine Scheduling Problem with not identical machines is presented. Specifically, a Reinforcement Learning Iterated Local Search meta-heuristic featuring a Q-learning Variable Neighbourhood Search is described to solve this problem and its performance is compared with the one of CPLEX solver. It is worthwhile mentioning that, for each of the proposed approaches, a thorough experimentation is performed and each Chapter is equipped with a detailed analysis of the results in order to appraise the performance of the method and to detect its limits

Efficient Algorithms for Scheduling Moldable Tasks

We study the problem of scheduling $n$ independent moldable tasks on $m$ processors that arises in large-scale parallel computations. When tasks are monotonic, the best known result is a $(\frac{3}{2}+\epsilon)$-approximation algorithm for makespan minimization with a complexity linear in $n$ and polynomial in $\log{m}$ and $\frac{1}{\epsilon}$ where $\epsilon$ is arbitrarily small. We propose a new perspective of the existing speedup models: the speedup of a task $T_{j}$ is linear when the number $p$ of assigned processors is small (up to a threshold $\delta_{j}$) while it presents monotonicity when $p$ ranges in $[\delta_{j}, k_{j}]$; the bound $k_{j}$ indicates an unacceptable overhead when parallelizing on too many processors. For a given integer $\delta\geq 5$, let $u=\left\lceil \sqrt[2]{\delta} \right\rceil-1$. In this paper, we propose a $\frac{1}{\theta(\delta)} (1+\epsilon)$-approximation algorithm for makespan minimization with a complexity $\mathcal{O}(n\log{\frac{n}{\epsilon}}\log{m})$ where $\theta(\delta) = \frac{u+1}{u+2}\left( 1- \frac{k}{m} \right)$ ($m\gg k$). As a by-product, we also propose a $\theta(\delta)$-approximation algorithm for throughput maximization with a common deadline with a complexity $\mathcal{O}(n^{2}\log{m})$