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New approximability results for 2dimensional packing problems
, 2006
"... The strip packing problem is to pack a set of rectangles into a strip of fixed width and minimum length. We present asymptotic polynomial time approximation schemes for this problem without and with 90 o rotations. The additive constant in the approximation ratios of both algorithms is 1, improvin ..."
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The strip packing problem is to pack a set of rectangles into a strip of fixed width and minimum length. We present asymptotic polynomial time approximation schemes for this problem without and with 90 o rotations. The additive constant in the approximation ratios of both algorithms is 1, improving on the additive term in the approximation ratios of the algorithm by Kenyon and Rémila (for the problem without rotations) and Jansen and van Stee (for the problem with rotations), both of which have a larger additive constant O(1/ε 2), ε> 0. The algorithms were derived from the study of the rectangle packing problem: Given a set R of rectangles with positive profits, the goal is to find and pack a maximum profit subset of R into a unit size square bin [0,1] × [0, 1]. We present algorithms that for any value ǫ> 0 find a subset R ′ ⊆ R of rectangles of total profit at least (1 − ǫ)OPT, where OPT is the profit of an optimum solution, and pack them (either without rotations or with 90 o rotations) into the augmented bin [0, 1] × [0, 1+ǫ].
A Structural Lemma in 2Dimensional Packing, and its Implications on Approximability
"... We present a new lemma stating that, given an arbitrary packing of a set of rectangles into a larger rectangle, a “structured” packing of nearly the same set of rectangles exists. This lemma has several implications on the approximability of 2dimensional packing problems. In this paper, we use it t ..."
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We present a new lemma stating that, given an arbitrary packing of a set of rectangles into a larger rectangle, a “structured” packing of nearly the same set of rectangles exists. This lemma has several implications on the approximability of 2dimensional packing problems. In this paper, we use it to show the existence of a polynomialtime approximation scheme for 2dimensional geometric knapsack when the range of the profit to area ratios of the rectangles is bounded by a constant. As a corollary, we get an approximation scheme for the problem of packing rectangles into a larger rectangle to occupy the maximum area. The existence of such an approximation scheme was a longstanding open problem, and has already been used in other papers on the absolute approximability of 2dimensional bin and strip packing. Moreover, we show that our approximation scheme can be used to find an asymptotic polynomialtime approximation scheme for 2dimensional fractional bin packing, the LP relaxation of the customary set covering formulation of 2dimensional bin packing. This also has already been used in another paper to improve the best known approximation guarantee for 2dimensional bin packing itself. The asymptotic approximation scheme is obtained by showing that the set covering LP relaxation can be modified slightly to obtain an almost equivalent LP, by introducing upper bounds on the dual LP variables, so that the dual separation problem reduces to the special case of 2dimensional geometric knapsack with bounded range of profit to area ratios mentioned above. We believe that this technique is of independent interest and should have other applications.
Approximation algorithms for orthogonal packing problems for hypercubes
 Theoretical Computer Science
, 2009
"... Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I = {r1,..., rn} of ddimensional rectangular items ri = (ai,1,..., ai,d) and a space Q. The task is t ..."
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Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I = {r1,..., rn} of ddimensional rectangular items ri = (ai,1,..., ai,d) and a space Q. The task is to pack the items in an orthogonal and nonoverlapping manner without using rotations into the given space. In the strip packing setting the space Q is given by a strip of bounded basis and unlimited height. The objective is to pack all items into a strip of minimal height. In the knapsack packing setting the given space Q is a single, usually unit sized bin and the items have associated profits pi. The goal is to maximize the profit of a selection of items that can be packed into the bin. We mainly focus on orthogonal knapsack packing restricted to hypercubes and our main results are a (5/4 + )approximation algorithm for twodimensional hypercube knapsack packing, also known as square packing, and a (1+1/2d+)approximation algorithm for ddimensional hypercube knapsack packing. In addition we consider ddimensional hypercube strip packing in the case of a bounded ratio between the shortest and longest side of the basis of the strip. We derive an asymptotic polynomial time approximation scheme (APTAS) for this problem. Finally, we present an algorithm that packs hypercubes with a total profit of at least (1 − )OPT into a large bin (the size of the bin depends on ). This problem is known as hypercube knapsack packing with large resources. A preliminary version was published in [15] but especially for the latter two approximation schemes no details were given due to page limitations.
Constant ratio approximation algorithms for weighted containerpacking
 InIPSJ SIG Technical Reports
, 2005
"... Given a set of rectangular threedimensional items, all of them associated with a profit, and a single bigger rectangular threedimensional bin, we can ask to find a nonrotational, nonoverlapping packing of a selection of these items into the bin to maximize the profit. This problem differs from t ..."
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Given a set of rectangular threedimensional items, all of them associated with a profit, and a single bigger rectangular threedimensional bin, we can ask to find a nonrotational, nonoverlapping packing of a selection of these items into the bin to maximize the profit. This problem differs from threedimensional strip and binpacking as we are to pack into a single bounded bin. We derive a (16 + )approximation algorithm and improve the algorithm to an approximation ratio of (9 + ). It turned out, that there is a mistake in Lemma 3.2 that affects the correctness of the (9+) algorithm. Instead of correcting the mistake here, we refer to [4]. Based on the ideas developed in this work, we derived a (9 + ) and a (8 + ) algorithm in [4]. 1
Weighted Rectangle and Cuboid Packing
, 2005
"... Given a set of rectangular items, all of them associated with a profit, and a single bigger rectangular bin, we can ask to find a nonrotational, nonoverlapping packing of a selection of these items into the bin to maximize the profit. A detailed description of the (2 + ɛ)approximation algorithm o ..."
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Given a set of rectangular items, all of them associated with a profit, and a single bigger rectangular bin, we can ask to find a nonrotational, nonoverlapping packing of a selection of these items into the bin to maximize the profit. A detailed description of the (2 + ɛ)approximation algorithm of Jansen and Zhang [8] for the two dimensional case is given. Furthermore we derive a (16 + ɛ)approximation algorithm for the threedimensional case (which we call cuboid packing) and improve this algorithm in a second step to an approximation ratio of (9 + ɛ). Finally we prove that cuboid packing does not admit an asymptotic PTAS. It turned out, that there is a mistake in Lemma 4.2 that affects the correctness of the (9 + ɛ) algorithm. Instead of correcting the mistake here, we refer to [4]. Based on the ideas developed in this work, we derived a (9+ɛ) and a (8+ɛ) algorithm in [4]. 1