Tight bounds for NF-based bounded-space online bin packing algorithms

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1/2. Using an NF-based online algorithm the authors proved an ACR of 13/9 = 1.44 … for any given buffer size not less than 1. They also gave a lower bound of 4/3 = 1.33 … for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243, and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of h∞ (r) for the r-parametric problem when the buffer capacity is 3. Since h∞ (2) = 1.42312 …, our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r. © Springer Science+Business Media, LLC 2017

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)

Online Two-Dimensional Load Balancing

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: - We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions. - If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions. - For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem

Hindsight Learning for MDPs with Exogenous Inputs

Many resource management problems require sequential decision-making under uncertainty, where the only uncertainty affecting the decision outcomes are exogenous variables outside the control of the decision-maker. We model these problems as Exo-MDPs (Markov Decision Processes with Exogenous Inputs) and design a class of data-efficient algorithms for them termed Hindsight Learning (HL). Our HL algorithms achieve data efficiency by leveraging a key insight: having samples of the exogenous variables, past decisions can be revisited in hindsight to infer counterfactual consequences that can accelerate policy improvements. We compare HL against classic baselines in the multi-secretary and airline revenue management problems. We also scale our algorithms to a business-critical cloud resource management problem -- allocating Virtual Machines (VMs) to physical machines, and simulate their performance with real datasets from a large public cloud provider. We find that HL algorithms outperform domain-specific heuristics, as well as state-of-the-art reinforcement learning methods.Comment: 53 pages, 6 figure

Machine Covering in the Random-Order Model

In the Online Machine Covering problem jobs, defined by their sizes, arrive one by one and have to be assigned to $m$ parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. In this work, we study the Machine Covering problem in the recently popular random-order model. Here no extra resources are present, but instead the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham's Greedy-strategy in this context and establish that its competitive ratio decreases slightly to $\Theta\left(\frac{m}{\log(m)}\right)$ which is asymptotically tight. Then, as our main result, we present an improved $\tilde{O}(\sqrt[4]{m})$-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound showing that no algorithm can have a competitive ratio of $O\left(\frac{\log(m)}{\log\log(m)}\right)$ in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest

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

Randomization can be as helpful as a glimpse of the future in online computation

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $\Omega(n)$ bits of advice to achieve a competitive ratio better than $H_k=\Omega(\log k)$, where $k$ is the cache size. Previously, it was only known that $\Omega(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $\Omega(n\log n)$ bits of advice. Previously, it was only known that $\Omega(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Ros\'en [ICALP'09]

Run Generation Revisited: What Goes Up May or May Not Come Down

In this paper, we revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M , and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline setting. (1) We analyze alternating-up-down replacement selection (runs alternate between sorted and reverse sorted), which was studied by Knuth as far back as 1963. We show that this simple policy is asymptotically optimal. Specifically, we show that alternating-up-down replacement selection is 2-competitive and no deterministic online algorithm can perform better. (2) We give online algorithms having smaller competitive ratios with resource augmentation. Specifically, we exhibit a deterministic algorithm that, when given a buffer of size 4M , is able to match or beat any optimal algorithm having a buffer of size M . Furthermore, we present a randomized online algorithm which is 7/4-competitive when given a buffer twice that of the optimal. (3) We demonstrate that performance can also be improved with a small amount of foresight. We give an algorithm, which is 3/2-competitive, with foreknowledge of the next 3M elements of the input stream. For the extreme case where all future elements are known, we design a PTAS for computing the optimal strategy a run generation algorithm must follow. (4) Finally, we present algorithms tailored for nearly sorted inputs which are guaranteed to have optimal solutions with sufficiently long runs