Scheduling in the Secretary Model

This paper studies online makespan minimization in the secretary model. Jobs, specified by their processing times, are presented in a uniformly random order. The input size n is known in advance. An online algorithm has to non-preemptively assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy Light Load [Albers and Hellwig, 2012] provides a very simple approach retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model

Polylogarithmic guarantees for generalized reordering buffer management

In the Generalized Reordering Buffer Management Problem (GRBM) a sequence of items located in a metric space arrives online, and has to be processed by a set of k servers moving within the space. In a single step the first b still unprocessed items from the sequence are accessible, and a scheduling strategy has to select an item and a server. Then the chosen item is processed by moving the chosen server to its location. The goal is to process all items while minimizing the total distance travelled by the servers. This problem was introduced in [Chan, Megow, Sitters, van Stee TCS 12] and has been subsequently studied in an online setting by [Azar, Englert, Gamzu, Kidron STACS 14]. The problem is a natural generalization of two very well-studied problems: the k-server problem for b=1 and the Reordering Buffer Management Problem (RBM) for k=1. In this paper we consider the GRBM problem on a uniform metric in the online version. We show how to obtain a competitive ratio of O(log k(log k+loglog b)) for this problem. Our result is a drastic improvement in the dependency on b compared to the previous best bound of O(√b log k), and is asymptotically optimal for constant k, because Ω(log k + loglog b) is a lower bound for GRBM on uniform metrics

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

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters