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

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 ?

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

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+?)-approximation algorithm for any constant ? > 0. We also achieve best-possible approximation factors for some relevant special cases

Mathematical models and decomposition methods for the multiple knapsack problem

We consider the multiple knapsack problem, that calls for the optimal assignment of a set of items, each having a profit and a weight, to a set of knapsacks, each having a maximum capacity. The problem has relevant managerial implications and is known to be very difficult to solve in practice for instances of realistic size. We review the main results from the literature, including a classical mathematical model and a number of improvement techniques. We then present two new pseudo-polynomial formulations, together with specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. Extensive computational experiments show the effectiveness of the proposed approaches

Enhanced pseudo-polynomial formulations for bin packing and cutting stock problems

We study pseudopolynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudopolynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly fewer constraints and variables than the classical models. We propose upper- and lower-bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present nontrivial adaptations of our techniques that solve two interesting problem variants, namely the variable-sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving on previous algorithms and finding several new proven optimal solutions