On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

Dynamic bin packing of unit fractions items

Cloud-based systems often face the problem of dispatching a stream of jobs to run on cloud servers in an online manner. Each job has a size that defines the resource demand for running the job. Each job is assigned to run on a cloud server upon its arrival and the job departs after it completes. The departure time of a job, however, is not known at the time of its arrival. Each cloud server has a fixed resource capacity and the total resource demand of all the jobs running on a server cannot exceed its capacity at all times. The objective of job dispatching is to minimize the total cost of the servers used, where the cost of renting each cloud server is proportional to its running hours by “pay-as-you-go” billing. The above job dispatching problem can be modeled as a variant of the dynamic bin packing (DBP) problem known as MinUsageTime DBP. In this paper, we study the competitiveness bounds of MinUsageTime DBP. We establish an improved lower bound on the competitive ratio of Any Fit family of packing algorithms, and a new upper bound of $\mu +3$ on the competitive ratio of the commonly used First Fit packing algorithm, where $\mu $ is the max/min job duration ratio. Our result significantly reduces the gap between the upper and lower bounds for the MinUsageTime DBP problem to a constant value independent of $\mu $ , and shows that First Fit packing is near optimal for MinUsageTime DBP.

In this paper, we study the problem of how to dispatch the play requests to the cloud servers in a cloud gaming system. We show that the dispatching strategy of play requests may heavily affect the total service cost of the cloud gaming system. The play request dispatching problem can be considered as a variant of the dynamic bin packing problem. However, we show that the classical bin packing algorithms such as First Fit and Best Fit are not efficient in dispatching the play requests in cloud gaming due to the diurnal workload pattern of online games. To address this issue, we propose an efficient request dispatching algorithm that assigns the play requests according to the predicted ending times of the game sessions. Through extensive evaluations using real online gaming traces, we show that the proposed dispatching algorithm can significantly reduce the resource waste of the cloud servers and thus decrease the total service cost compared to the First Fit and Best Fit algorithms.

Partially dynamic bin packing can be solved within 1 + ε in (amortized) polylogarithmic time

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

Approximation and online algorithms for multidimensional bin packing: A survey

Online algorithms are a rich area of research with widespread applications in scheduling, combinatorial optimization, and resource allocation problems. This lucid textbook provides an easy but rigorous introduction to online algorithms for graduate and senior undergraduate students. In-depth coverage of most of the important topics is presented with special emphasis on elegant analysis. The book starts with classical online paradigms like the ski-rental, paging, list-accessing, bin packing, where performance of online algorithms is studied under the worst-case input and moves on to newer paradigms like 'beyond worst case', where online algorithms are augmented with predictions using machine learning algorithms. The book goes on to cover multiple applied problems such as routing in communication networks, server provisioning in cloud systems, communication with energy harvested from renewable sources, and sub-modular partitioning. Finally, a wide range of solved examples and practice exercises are included, allowing hands-on exposure to the concepts.

On-line algorithms for 2-space bounded 2-dimensional bin packing

The 2-dimensional bin packing problem (2BP) is a generalization of the classical Bin Packing problem and is defined as follows: Given a collection of rectangles specified by their width and height, pack these into the minimum number of square bins of unit size. We study the case of 'orthogonal packing without rotations', where rectangles cannot be rotated and must be packed parallel to the edges of a bin. Often in practical cases of 2BP problems there are additional constraints on how complicated the packing patterns in a bin can be. A well-studied and frequently used constraint is that every rectangle in the packing must be obtainable by recursively applying a sequence of edge-to-edge cuts parallel to the edges of the bin. Such cuts are known as guillotine cuts. Our main result is that the guillotine 2BP problem admits an asymptotic polynomial time approximation scheme. This is in sharp contrast with the fact that the general 2BP problem is APX-Hard. En route to our main result, we show a structural theorem about approximating general guillotine packings by simpler packings, which could be of independent interest.

Bin Packing Problem (BPP) is a problem that aims to minimize the number of container usage by maximizing its contents. BPP can be applied to a case, such as maximizing the printing of a number of stickers on a sheet of paper of a certain size. Genetic Algorithm is one way to overcome BPP problems. Examples of the use of a combination of BPP and Genetic Algorithms are applied to printed paper in Digital Printing companies. Genetic Algorithms adopt evolutionary characteristics, such as selection, crossover and mutation. Repeatedly, Genetic Algorithms produce individuals who represent solutions. However, this algorithm often does not achieve maximum results because it is trapped in a local search and a case of premature convergence. The best results obtained are not comprehensive, so it is necessary to modify the parameters to improve this condition. Random Crossover and Dynamic Mutation were chosen to improve the performance of Genetic Algorithms. With this application, the performance of the Genetic Algorithm in the case of BPP can overcome premature convergence and maximize the allocation of printing and the use of paper. The test results show that an average of 99 stickers can be loaded on A3 + size paper and the best generation is obtained on average in the 21 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">st</sup> generation and the remaining space is 3,500 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .

A Computational Study of Lower Bounds for the Two Dimensional Bin Packing Problem

Abstract. We consider the on-line Dual Bin Packing problem where we have nunit size bins and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. We investigate unrestrictedalgorithms that have the power of performing admission control on the items, i.e., rejecting items while there is enough space to pack them, versus fairalgorithms that reject an item only when there is not enough space to pack it. We show that by performing admission control on the items, we get better performance compared with the performance achieved on the fair version of the problem. Our main result shows that with an unfair variant of First-Fit, we can pack approximately two-thirds of the items for sequences for which an optimal off-line algorithm can pack all the items. This is in contrast to standard First-Fit where we show an asymptotically tight hardness result: if the number of bins can be chosen arbitrarily large, the fraction of the items packed by First-Fit comes arbitrarily close to five-eighths.

Online makespan minimization is a classical problem in which a sequence of jobs $$\sigma = J_1, \ldots , J_n$$ź=J1,ź,Jn has to be scheduled on m identical parallel machines so as to minimize the maximum completion time of any job. In this paper we investigate the problem with an essentially new model of resource augmentation. More specifically, an online algorithm is allowed to build several schedules in parallel while processing $$\sigma $$ź. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. As a main result we develop a $$(4/3+\varepsilon )$$(4/3+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that uses a constant number of schedules. The constant is equal to $$1/\varepsilon ^{O(\log (1/\varepsilon ))}$$1/źO(log(1/ź)). We also give a $$(1+\varepsilon )$$(1+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that builds a polynomial number of $$(m/\varepsilon )^{O(\log (1/\varepsilon ) / \varepsilon )}$$(m/ź)O(log(1/ź)/ź) schedules. This value depends on m but is independent of the input $$\sigma $$ź. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4 / 3 must construct $$\Omega (m)$$Ω(m) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence $$\sigma $$ź to within a factor of $$1+\varepsilon $$1+ź and (2) guess the job processing times and their frequencies in $$\sigma $$ź. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. We remark that the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation. Furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization.

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s1,..., s ∈ (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j′) ∈ E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem. We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ∈) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in $$\tfrac{1}{\varepsilon}$$ .

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.