Online Strip Packing with Polynomial Migration

We consider the relaxed online strip packing problem, where rectangular items arrive online and have to be packed into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio 1 + O(epsilon) for any epsilon > 0 and amortized migration factor polynomial in 1/epsilon. To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}{\epsilon}$. This answers an open question stated by Epstein and Levin in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. for linear programs

Online Bin Covering with Limited Migration

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor beta: When an object o of size s(o) arrives, the decisions for objects of total size at most beta * s(o) may be revoked. Usually beta should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classical problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small epsilon). We therefore resolve the competitiveness of the bin covering problem with migration

Multiple Strip Packing and Scheduling Parallel Jobs in Platforms

We consider two strongly related problems, multiple strip packing and scheduling parallel jobs in platforms. In the first one we are given a list of n rectangles with heights and widths bounded by one and N strips of unit width and infinite height. The objective is to find a non-overlapping orthogonal packing without rotations of all rectangles into the strips minimizing the maximum height used. In the scheduling problem we consider jobs instead of rectangles, i.e. we are allowed to cut the rectangles vertically and we may have target areas of different size, called platforms. A platform $P_\ell$ is a collection of $m_\ell$ processors running at speed $s_\ell$ and the objective is to minimize the makespan, i.e. the latest finishing time of a job

Cardinality Constrained Scheduling in Online Models

Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham's famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most $k$ jobs to each machine where $k$ is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of $2$ on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size $p$, we are allowed to migrate jobs of total size at most a constant times $p$. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches.Comment: An extended abstract will appear in the proceedings of STACS'2

Approximation algorithms for scheduling and two-dimensional packing problems

This dissertation thesis is concerned with two topics of combinatorial optimization : scheduling and geometrical packing problems. Scheduling deals with the assignment of jobs to machines in a ‘good’ way, for suitable notions of good. Two particular problems are studied in depth : on the one hand, we consider the impact of machine failure on online scheduling, i.e. what are the consequences of the fact that in real life, machines do not work flawlessly around the clock, but need maintenance intervals or can break down? How do we need to adapt our algorithms to still obtain good overall schedules, and in what settings do we even have a chance to succeed? Our second problem is of a more static nature : in some settings, not every job is permitted on all the machines. A classical example would be that of workers which needs special qualification to execute some jobs, or a certain minimum requirement of memory size of computers, etc. The problem in general is notoriously hard to tackle; we present improved approximation ratios for several special cases. In particular, we derive a polynomial-time approximation scheme for nested interval restrictions, which occur naturally in many practical applications. Our final topic is two-dimensional geometric bin packing, the problem of packing rectangular objects into the minimum number of containers of identical size (figuratively speaking, we are arranging advertisements of fixed dimensions into the minimum number of print pages). It is known that no approximation ratio better than 2 is possible for this problem, unless P = NP; we present an algorithm that guarantees this ratio.Diese Promotionsschrift behandelt zwei Arten kombinatorischer Optimierungsprobleme : Ablaufplanungsprobleme und geometrische Packungsprobleme. Ablaufplanungsprobleme handeln davon, eine Menge von Aufgaben, die Jobs, auf eine Menge von ausführenden Maschinen oder Arbeitern zu verteilen, so dass der entstehende Ablaufplan in geeignetem Sinne gut ist. Wir betrachten hier insbesondere folgende zwei Probleme der Ablaufplanung: einerseits untersuchen wir den Einfluß von Maschinenausfällen auf die Online-Ablaufplanung: im wirklichen Leben sind Maschinen nicht fehler- und unterbrechungslos verfügbar. Wir geben eine teilweise Antwort auf die Frage, mit welchen Änderungen Algorithmen trotz unerwartet auftretender Maschinenausfälle gute Pläne erstellen können, und in welchen Fällen es prinzipiell nicht möglich ist, gute Ablaufpläne zu erstellen. Unser zweites Ablaufplanungsproblem ist von statischerer Natur: in der praktischen Anwendung ist es häufig der Fall, dass nicht jede Maschine jeden Job ausführen kann. Ein einfaches Beispiel sind menschliche Arbeiter, die gewisse formale Qualifikationen für gewisse Jobs haben müssen. Diese Problem erweist sich als in voller Allgemeinheit bekannt hartnäckig; wir stellen hier Algorithmen für einige Spezialfälle vor. Insbesondere präsentieren wir ein polynomielles Approximationsschema für den wichtigen Fall verschachtelter Restriktionen, der in der Mitarbeiterplanung auf natürliche Weise auftritt. Schlussendlich untersuchen wir das zweidimensionale geometrische bin packing-Problem. Fragestellung dieses Problem ist es, rechteckige Objekte in die minimale Anzahl von Containern gleicher Größe zu packen. Salopp gesprochen versuchen wir, eine vorgegebene Menge von Anzeigen mit vorgegebenen Abmessungen auf eine möglichst kleine Zahl von Druckseiten gleicher Größe zu platzieren. Es ist bekannt, dass dieses Problem keine Algorithmus mit Approximationsgüte besser als 2 erlaubt, es sei denn, P = NP; wir stellen einen Algorithmus mit Güte 2 vor