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Bin packing in multiple dimensions: Inapproximability results and approximation schemes
 MATHEMATICS OF OPERATIONS RESEARCH
, 2006
"... We study the multidimensional generalization of the classical Bin Packing problem: Given a collection of ddimensional rectangles of specified sizes, the goal is to pack them into the minimum number of unit cubes. A long history of results exists for this problem and its special cases. Currently, t ..."
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We study the multidimensional generalization of the classical Bin Packing problem: Given a collection of ddimensional rectangles of specified sizes, the goal is to pack them into the minimum number of unit cubes. A long history of results exists for this problem and its special cases. Currently, the best known approximation algorithm for packing twodimensional rectangles achieves a guarantee of 1.69 in the asymptotic case (i.e., when the optimum uses a large number of bins) [3]. An important open question has been whether 2−dimensional bin packing is essentially similar to the 1−dimensional case in that it admits an asymptotic polynomial time approximation scheme (APTAS) [12, 17] or not. We answer the question in the negative and show that the problem is APX hard in the asymptotic sense. On the positive side, we give the following results: First, we consider the special case where we have to pack ddimensional cubes into the minimum number of unit cubes. We give an asymptotic polynomial time approximation scheme for this problem. This represents a significant improvement over the previous best known asymptotic approximation factor of 2 − (2/3) d [21] (1.45 for d = 2 [11]), and settles the approximability of the problem. Second, we give a polynomial time algorithm for packing arbitrary rectangles into at most OPT square bins with sides of length 1 + ε, where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result does not have an additive constant term i.e., is not an asymptotic result. As a corollary, we obtain a polynomial time approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle, settling also the approximability of this problem.
On packing squares with resource augmentation: maximizing the profit
 In Proc. of the Australasian Computer Science Week, Computing: The Australasian Theory Symposium (CATS ’05
, 2005
"... We consider the problem of packing squares with profits into a bounded square region so as to maximize their total profit. More specifically, given a set L of n squares with positive profits, it is required to pack a subset of them into a unit size square region�0�1℄¢�0�1℄so that the total profit of ..."
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We consider the problem of packing squares with profits into a bounded square region so as to maximize their total profit. More specifically, given a set L of n squares with positive profits, it is required to pack a subset of them into a unit size square region�0�1℄¢�0�1℄so that the total profit of the squares packed is maximized. For any given positive accuracy ε�0, we present an algorithm that outputs a packing of a subset of L in the augmented square region �1 ε℄¢�1 ε℄with profit value at least 1 ε OPT L, where OPT L is the maximum profit that can be achieved by packing a subset of L in the unit square. The running time of the algorithm is polynomial in n for fixed ε.
Online squareintosquare packing
, 2014
"... In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects ..."
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In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself. We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to accommodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a 2.82...competitive method for minimizing the required container size, and a lower bound of 1.33... for the achievable factor.
Approximation Schemes for Multidimensional Packing
, 2003
"... We consider a classic multidimensional generalization of the bin packing problem, namely, packing ddimensional rectangles into the minimum number of unit cubes. Our two results are: an asymptotic polynomial time approximation scheme for packing d dimensional cubes into the minimum number of unit c ..."
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We consider a classic multidimensional generalization of the bin packing problem, namely, packing ddimensional rectangles into the minimum number of unit cubes. Our two results are: an asymptotic polynomial time approximation scheme for packing d dimensional cubes into the minimum number of unit cubes and a polynomial time algorithm for packing rectangles into at most OPT bins whose sides have length (1 + #), where OPT denotes the minimum number of unit bins required to pack the rectangles. Both algorithms also achieve the best possible additive constant term. For cubes, this settles the approximability of the problem and represents a significant improvement over the previous best known asymptotic approximation factor of 2 + #. For rectangles, this contrasts with the currently best known approximation factor of 1.691 . . ..
On Packing Rectangles with Resource Augmentation: Maximizing the Profit
 ALGORITHMIC OPERATIONS RESEARCH VOL.3 (2008) 1–12
, 2008
"... We consider the problem of packing rectangles with profits into a bounded square region so as to maximize their total profit. More specifically, given a set R of n rectangles with positive profits, it is required to pack a subset of them into a unit size square frame [0,1] × [0,1] so that the total ..."
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We consider the problem of packing rectangles with profits into a bounded square region so as to maximize their total profit. More specifically, given a set R of n rectangles with positive profits, it is required to pack a subset of them into a unit size square frame [0,1] × [0,1] so that the total profit of the rectangles packed is maximized. For any given positive accuracy ε> 0, we present an algorithm that outputs a packing of a subset of R in the augmented square region [1+ε] × [1+ε] with profit value at least (1−ε)OPT, where OPT is the maximum profit that can be achieved by packing a subset of R in a unit square frame. The running time of the algorithm is polynomial in n for fixed ε.