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

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

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 ?

High Multiplicity Strip Packing Problem With Three Rectangle Types

The two-dimensional strip packing problem (2D-SPP) involves packing a set R = {r1, ..., rn} of n rectangular items into a strip of width 1 and unbounded height, where each rectangular item ri has width 0 \u3c wi ≤ 1 and height 0 \u3c hi ≤ 1. The objective is to find a packing for all these items, without overlaps or rotations, that minimizes the total height of the strip used. 2D-SPP is strongly NP-hard and has practical applications including stock cutting, scheduling, and reducing peak power demand in smart-grids. This thesis considers a special case of 2D-SPP in which the set of rectangular items R has three distinct rectangle sizes or types. We present a new OPT + 5/3 polynomial-time approximation algorithm, where OPT is the value of an optimum solution. This algorithm is an improvement over the previously best OPT + 2 polynomial-time approximation algorithm for the problem

Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

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

Verbesserte Approximationsalgorithmen für Packungs- und Ablaufplanungsprobleme

This thesis presents approximation algorithms for geometric packing and scheduling problems. First, improved AFPTAS for the Bin Packing Problem (BP) and its generalization, the Variable-sized Bin Packing Problem (VBP), are explained. Our algorithms have to solve the unbounded variant of the Knapsack Problem (KP) and of the Knapsack Problem with Inversely Proportional Profits (KPIP) as subproblems. In the normal 0-1 variant of KP, an item can be chosen only once. In the bounded variant, an individual bounded number of copies can be taken of every item. The unbounded variant (UKP) allows for an infinite number of copies of every item. KPIP is a generalization of KP in which we have not only one, but several knapsack sizes. The profit of an item is inversely proportional to the size of the knapsack into which it has been packed. This makes it non-trivial to choose the knapsack size that maximizes the profit over all knapsack sizes. Similar to KP, there are the 0-1, the bounded, and the unbounded variant of KPIP. We first present FPTAS for every of the three variants of KPIP. They are faster than the natural approach to separately solve for every knapsack size the corresponding Knapsack Problem. Second, we present an FPTAS for UKP that is faster and needs less storage space than previously known algorithms. Finally, we combine the approaches of the KPIP and of the UKP FPTAS to get an FPTAS for the Unbounded KPIP that has again a better time and space complexity. All these results improve the running time for our BP and VBP algorithms. As a corollary, we also improve the running time for a Strip Packing AFPTAS. Finally, we consider Scheduling on Unrelated Machines of which we study the special case with a constant number K of machine types: one job has the same processing time on every machine of the same type. We present a PTAS for this special case. The algorithm has a better running time than the previously known algorithm for general (but constant) K.Diese Dissertation stellt Approximationsalgorithmen für geometrische Packungs- und Ablaufplanungsprobleme (Packing and Scheduling Problems) vor. Zuerst werden verbesserte AFPTAS für das Behälterproblem (Bin Packing, BP) und seine Verallgemeinerung, das Behälterproblem mit verschiedenen Behältergrößen (Variable-sized Bin Packing, VBP), erklärt. Unsere Algorithmen müssen die unbeschränkte (unbounded) Variante des Rucksackproblems (Knapsack Problem, KP) und des Rucksackproblems mit invers proportionalen Profiten (Knapsack Problem with Inversely Proportional Profits, KPIP) als Unterprobleme lösen. Bei KP gibt es die Varianten 0-1, beschränkt und unbeschränkt. KPIP ist eine Verallgemeinerung des Rucksackproblems mit mehreren Rucksackgrößen, die in dieser Dissertation eingeführt wird. Wie bei KP gibt es bei KPIP ebenfalls die Varianten 0-1, beschränkt und unbeschränkt. Wir stellen zuerst FPTAS für alle drei Varianten von KPIP vor. Sie sind schneller als der natürliche Ansatz, für jede Rucksackgröße das entsprechende Rucksackproblem einzeln zu lösen. Danach stellen wir ein FPTAS für UKP vor, das schneller ist und weniger Speicherplatz benötigt als zuvor bekannte Algorithmen. Schließlich kombinieren wir den Ansatz für KPIP und für das unbeschränkte KPIP, um ein FPTAS für die unbeschränkte Variante von KPIP zu erhalten, das wiederum eine kleinere Zeit- und Speicherkomplexität besitzt. All diese Resultate verbessern die Laufzeit unserer BP- und VBP-Algorithmen. Als Korollar verbessern wir außerdem die Laufzeit eines AFPTAS für das geometrische Zuschnittproblem (Strip Packing). Schließlich betrachten wir das Ablaufplanungsproblem auf heterogenen Maschinen (Scheduling on Unrelated Machines), bei dem wir den Spezialfall mit einer konstanten Anzahl K an Maschinentypen untersuchen: Eine Aufgabe (Job) hat auf jeder Maschine desselben Typs die gleiche Ausführungszeit. Wir stellen für diesen Spezialfall ein PTAS vor. Der Algorithmus ist schneller als das zuvor bekannte Verfahren für allgemeines (aber konstantes) K