Probabilistic analysis of algorithms for dual bin packing problems
TL;DR: A probabilistic analysis of the dual bin packing problem is carried out under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and the connection between this problem and the classical binpacking problem as well as to renewal theory is revealed.
About: This article is published in Journal of Algorithms.The article was published on 1991-04-01 and is currently open access. It has received 43 citations till now. The article focuses on the topics: Bin packing problem & Bin.
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TL;DR: This survey considers approximation and online algorithms for several classical generalizations of bin packing problem such as geometric bin packing, vector bin packing and various other related problems.
202 citations
Proceedings Article•
01 Jan 1999TL;DR: Packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height.
Abstract: We study the approximability of multi-dimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n d-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension accross all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems. Keywords Multi-dimensional packing, vector scheduling, vector bin packing, packing integer programs, multiprocessor scheduling, bin packing, knapsack, approximation algorithms, hardness of approximation, combinatorial optimization Disciplines Computer Engineering Comments Postprint version. Copyright SIAM, 2004. Published in SIAM Journal on Computing, Volume 33, Issue 4, 2004, pages 837-851. Publisher URL: http://dx.doi.org/10.1137/S0097539799356265 This journal article is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/78 ! #"$ % &' )(* ,+ . 0/ 1320465#798:4;1329 =*?98A@CB D#465FEA !fFg hGn plYAk
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200 citations
184 citations
Cites background or methods from "Probabilistic analysis of algorithm..."
...(Csirik, Frenk, Galambos , R i n n o o y Kan [33]) If the items in L,~ are distributed according to U(0, a), 0 < a < 1, then E[DNF(Ln)] grows like n / v where...
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...Several years later, Csirik, Frenk, Galambos and Rinnooy Kan [33] showed that the Y2(n 1/2) in this formula is in fact @(nl/2)....
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...The obvious extension DNF2 of the DUAL NEXT FIT algorithm to 2-dimensional vector covering was analyzed by Csirik et al [33]....
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Posted Content•
20 Feb 2006
TL;DR: A list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005 can be found in this article.
Abstract: This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.
84 citations
09 Jan 2001
TL;DR: This work designs algorithms that have bounded worst-case behavior for instances with discrete item sizes and expected behavior that is asymptotically optimal for all discrete “perfect-packing distributions” (ones for which optimal packings have sublinear expected waste).
Abstract: Bin covering takes as input a list of items with sizes in (0, 1) and places them into bins of unit demand so as to maximize the number of bins whose demand is satisfied. This is in a sense a dual problem to the classical one-dimensional bin packing problem, but has for many years lagged behind the latter in terms of the quality of the best approximation algorithms. We design algorithms for this problem that close the gap, both in terms of worst- and average-case results. We present (1) the first asymptotic approximation scheme for the offline version, (2) algorithms that have bounded worst-case behavior for instances with discrete item sizes and expected behavior that is asymptotically optimal for all discrete “perfect-packing distributions” (ones for which optimal packings have sublinear expected waste), and (3) a learning algorithm that has asymptotically optimal expected behavior for all discrete distributions. The algorithms of (2) and (3) are based on the recently-developed online Sum-of-Squares algorithm for bin packing. We also present experimental analysis comparing the algorithms of (2) and suggesting that one of them, the Sum-of-Squares-with-Threshold algorithm, performs quite well even for discrete distributions that do not have the perfect-packing property.
67 citations
Cites result from "Probabilistic analysis of algorithm..."
...Previous work on average-case analysis for bin covering either deMt only with the convergence properties of A(Ln(F))/n (for arbitrary distributions) [14, 17, 18], or the value of ERAs(F) for the particular distribution U[0, 1] in which item sizes are uniformly distributed in the real interval [0, 1], as in [ 8 ]....
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References
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Book•
01 Jan 2001
TL;DR: This edition of A Course in Probability Theory includes an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.
Abstract: Since the publication of the first edition of this classic textbook over thirty years ago, tens of thousands of students have used A Course in Probability Theory. New in this edition is an introduction to measure theory that expands the market, as this treatment is more consistent with current courses. While there are several books on probability, Chung's book is considered a classic, original work in probability theory due to its elite level of sophistication.
2,647 citations
01 Jan 1975
236 citations
TL;DR: Approximation algorithms are presented that provide guarantees of 1 2, 2 3 , and 3 4 the optimal number, at running time costs of O(n), O(nlogn), and O( nlog2n), respectively, and the average case behavior of these algorithms is explored via empirical tests on randomly generated sets of items.
176 citations