We study online multidimensional variants of the generalized assignment problem which are used to model prominent real-world applications, such as the assignment of virtual machines with multiple resource requirements to physical infrastructure in cloud computing. These problems can be seen as an extension of the well known secretary problem and thus the standard online worst-case model cannot provide any performance guarantee. The prevailing model in this case is the random-order model, which provides a useful realistic and robust alternative. Using this model, we study the @math -dimensional generalized assignment problem, where we introduce a novel technique that achieves an @math -competitive algorithms and prove a matching lower bound of @math . Furthermore, our algorithm improves upon the best-known competitive-ratio for the online (one-dimensional) generalized assignment problem and the online knapsack problem.
Some related problems have competitive algorithms in the worst-case model too. One example is the AdWords problem which is a special case of GAP in which the profit of each item is equal to its size. Under the assumption that items are small compared to the capacity of the bins, @cite_6 presented an optimal @math -competitive algorithm. Without this assumption, the best known competitive-ratio is @math @cite_0 . Another example is the online vector bin packing problem, in which items arrive one-by-one, and the goal is to pack them all in the minimum number of unit sized @math -dimensional bins. This problem was studied by @cite_8 who showed that the First Fit algorithm has a worst-case competitive-ratio of @math . More recently, @cite_19 showed that this algorithm is asymptotically optimal by proving a lower bound of @math .
@cite_0 @cite_19 @cite_6 @cite_8
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We revisit the classic online bin packing problem studied in the half-century. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called bins of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class of items to be packed in multiple kinds of bins, and not only into their own bins. We combine this with the approach of packing of sufficiently big items according to their exact sizes. Finally, we simplify the analysis of such algorithms, allowing the analysis to be based on the most standard weight functions. This simplified analysis allows us to study the algorithm which we defined based on all these ideas. This leads us to the design and analysis of the first algorithm of asymptotic competitive ratio strictly below 1.58, specifically, we break this barrier by providing an algorithm AH (Advanced Harmonic) whose asymptotic competitive ratio does not exceed 1.57829. Our main contribution is the introduction of the simple analysis based on weight function to analyze the state of the art online algorithms for the classic online bin packing problem. The previously used analytic tool named weight system was too complicated for the community in this area to adjust it for other problems and other algorithmic tools that are needed in order to improve the current best algorithms. We show that the weight system based analysis is not needed for the analysis of the current algorithms for the classic online bin packing problem. The importance of a simple analysis is demonstrated by analyzing several new features together with all existing techniques, and by proving a better competitive ratio than the previously best one. We study randomized algorithms for the online vector bin packing and vector scheduling problems. For vector bin packing, we achieve a competitive ratio of O(d1 B), where d is the number of dimensions and B the size of a bin. This improves the previous bound of O(d1 (B−1)) by a polynomial factor, and is tight up to logarithmic factors. For vector scheduling, we show a lower bound of [EQUATION] on the competitive ratio of randomized algorithms, which is the first result for randomized algorithms and is asymptotically tight. Finally, we analyze the widely used "power of two choices' algorithm for vector scheduling, and show that its competitive ratio is [EQUATION], which is optimal up to the additive O(log log n) term that also appears in the scalar version of this algorithm. The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit. We study a variant of online bin packing, called colorful bin packing. In this problem, items that are presented one by one are to be packed into bins of size 1. Each item i has a size s i ∈ [0,1] and a color (c_i C ), where ( C ) is a set of colors (that is not necessarily known in advance). The total size of items packed into a bin cannot exceed its size, thus an item i can always be packed into a new bin, but an item cannot be packed into a non-empty bin if the previous item packed into that bin has the same color, or if the occupied space in it is larger than 1 − s i . This problem generalizes standard online bin packing and online black and white bin packing (where (| C |=2 )). We prove that colorful bin packing is harder than black and white bin packing in the sense that an online algorithm for zero size items that packs the input into the smallest possible number of bins cannot exist for (| C | 3 ), while it is known that such an algorithm exists for (| C |=2 ). We show that natural generalizations of classic algorithms for bin packing fail to work for the case (| C | 3 ), and moreover, algorithms that perform well for black and white bin packing do not perform well either, already for the case (| C |=3 ). Our main results are a new algorithm for colorful bin packing that we design and analyze, whose absolute competitive ratio is 4, and a new lower bound of 2 on the asymptotic competitive ratio of any algorithm, that is valid even for black and white bin packing.