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 load balancing with general reassignment cost

We investigate a semi-online variant of load balancing with restricted assignment. In this problem, we are given n jobs, which need to be processed by m machines with the goal to minimize the maximum machine load. Since strong lower bounds rule out any competitive ratio of o(log⁡n), we may reassign jobs at a certain job-individual cost. We generalize a result by Gupta, Kumar, and Stein (SODA 2014) by giving a O(log⁡log⁡mn)-competitive algorithm with constant amortized reassignment cost

Fully Dynamic Bin Packing Revisited

We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ration of $1 + \epsilon$ for the number of bins, a relatively simple argument proves a lower bound of $\Omega(\frac{1}{\epsilon})$ for the migration factor. We establish a nearly matching upper bound of $O(\frac{1}{\epsilon}^4 \log \frac{1}{\epsilon})$ using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items $n$ and in $\frac{1}{\epsilon}$. The previous best trade-off was for an asymptotic competitive ratio of $\frac{5}{4}$ for the bins (rather than $1+\epsilon$) and needed an amortized number of $O(\log n)$ repackings (while in our scheme the number of repackings is independent of $n$ and non-amortized)

Über Struktur- und Sensitivitätsaussagen in Ganzzahligen Programmen und deren Anwendung in Kombinatorischer Optimierung

In this thesis we investigate properties of integer linear programs (ILPs) and their algorithmic use. Our main focus are ILP-formulations that come from concrete algorthmic problems like the bin packing problem or the scheduling problem on identical machines. Especially for this kind of ILPs we study structural properties as well as properties for their sensitivity. As a result, we are able to answer open algorithmical questions in the area of approximation and parameterized complexity. In the context of sensitivity we analyze how much an ILP solution has to be adjusted when the parameters of the ILP change. There is a classical results by Cook et al. which gave bounds for that question when optimal solutions are considered. However, in this thesis we investigate the sensitivity of ILPs when approximate solutions are allowed, i.e. solutions that differ by a factor of at most (1+ \epsilon) from the optimum value. We could apply the obtained results to the online bin packing problem, when an approximation guarantee with ratio $1+ \epsilon$ has to be fulfilled and repacking of already assigned items (limited by the so called migration factor) is allowed. In the context of structural results, we prove the existence (assuming the ILP is feasible) of solutions of a certain class of ILPs with a certain simplified structure. Specifically, in this thesis, we prove structure properties for ILPs that arise from formulations of bin packing or scheduling problems and natural generalization of those formulations. Based on the those structure properties, we develop an efficient approximation scheme for the scheduling problem on identical machines with a running time of 2^{\tilde{O}(1/\epsilon)} + poly}(n) and furthermore, we develop a structure theorem, which is applied to the bin packing problem when the number of different item sizes d is bounded.In dieser Dissertation werden Eigenschaften von ganzzahligen linearen Programmen (engl. integer linear programs, kurz: ILPs) untersucht. Von Interesse sind dabei hauptsächlich ILP-Formulierungen, welche sich aus dem Kontext von algorithmischen Problemstellungen ergeben, wie beispielsweise dem Bin Packing-Problem und dem Scheduling-Problem auf identischen Maschinen. Insbesondere für diese ILPs zeigen wir Strukturaussagen, sowie Aussagen über die Sensitivität und können so offene algorithmische Fragestellungen im Bereich von Approximation und parametrisierter Komplexität lösen. Im Kontext von Sensitivitätsaussagen wird untersucht, inwiefern Lösung des ILPs angepasst werden können, wenn sich die Parameter des ILPs leicht ändern. Ein klassisches Resultat von Cook u.a. gibt dabei für optimale Lösungen des ILPs Abschätzungen an. In dieser Arbeit betrachten wir Abschätzungen für die Senstivität wenn approximative Lösungen erlaubt sind, d.h. Lösungen deren Zielfunktionswert höchstens um einen Faktor 1+ \epsilon über dem optimalen Zielfunktionswert liegt. Diese Ergebnisse konnten wir auf das Online-Bin Packing-Problem anwenden, wenn eine approximative Lösung mit Güte 1+ \epsilon erreicht werden soll und in beschränktem Maße Items umgepackt werden dürfen. Im Kontext von Strukturaussagen wird in dieser Dissertation die Existenz von ILP-Lösungen bewiesen, welche eine bestimmte vereinfachte Struktur aufweisen. Insbesondere, konnten wir Strukturaussagen für ILPs entwickeln, welche sich aus Formulierungen des Bin Packing-Problems ergeben bzw. natürliche Verallgemeinerungen dieser Formulierung. Dadurch ist es uns zum einen gelungen ein effizientes Approximationsschemata für das Scheduling-Problem auf identischen Maschinen mit einer Laufzeit von 2^{\tilde{O}(1/\epsilon)} + poly(n) zu entwicklen und außerdem konnten wir eine Strukturaussage entwickeln, welche unter anderem Anwendung im Bin Packung-Problem fand, wenn die Anzahl der unterschiedlichen Itemgrößen d beschränkt ist