Rectangle packing with one-dimensional resource augmentation

Jansen, Klaus; Solis-Oba, Roberto

doi:10.1016/j.disopt.2009.04.001

Cited by 42 publications

(45 citation statements)

References 20 publications

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“…Since both the problem with and without rotations are not approximable to any 2 − unless P = N P , this settles the question of absolute approximability of these problems. For practical applications, it would be interesting to find faster algorithms: our algorithm relies heavily on the knapsack PTAS in [3] and techniques in [14] with a doubly-exponential dependency on , in particular when compared to the running time O(n log n) of Harren and van Stee's 3-approximation in [10].…”

Section: Resultsmentioning

confidence: 99%

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Two for One: Tight Approximation of 2D Bin Packing

Jansen

Prädel

Schwarz

2009

Lecture Notes in Computer Science

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In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of unit bins without rotating the rectangles. Beyond its theoretical appeal, this problem has many practical applications, for example in print layout and VLSI chip design.We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the problem where rotations are allowed and also matches the known lower bound of approximability. Our approach makes strong use of a PTAS for a related 2D knapsack problem and a new algorithm that can pack instances into two bins if OPT = 1.

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Section: Resultsmentioning

confidence: 99%

“…In recent years, there has been increasing interest in extensions of packing problems such as strip packing [1,2,14,17,19], knapsack [3,15,18] and bin packing [4,5,6,9,20], to multiple independent criteria (vector packing) or multiple dimensions (geometric packing).…”

Section: Introductionmentioning

confidence: 99%

Two for One: Tight Approximation of 2D Bin Packing

Jansen

Prädel

Schwarz

2009

Lecture Notes in Computer Science

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“…An asymptotic FPTAS with additive constant of O(h max (I)/ε 2 ) was given by Kenyon & Rémila [12]. Jansen & Solis-Oba [9] found an asymptotic PTAS with additive constant of h max (I).…”

Section: A(i)mentioning

confidence: 99%

“…Two-dimensional packing problems are classical in combinatorial optimization and continue to receive a lot of research interest [4,5,8,14,9,11,12]. One of the most important ones is the strip packing problem also known as the cutting stock problem: given a set of rectangles I = {r 1 , .…”

Section: Introductionmentioning

confidence: 99%

A (5/3 + ε)-Approximation for Strip Packing

Harren

Jansen

Prädel

et al. 2011

Lecture Notes in Computer Science

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We study strip packing, which is one of the most classical two-dimensional packing problems: given a collection of rectangles, the problem is to find a feasible orthogonal packing without rotations into a strip of width 1 and minimum height. In this paper we present an approximation algorithm for the strip packing problem with absolute approximation ratio of 5/3 + ε for any ε > 0. This result significantly narrows the gap between the best known upper bounds of 2 by Schiermeyer and Steinberg and 1.9396 by Harren and van Stee and the lower bound 3/2.

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“…For example, in Section 5 we obtain several results related to resource augmented packing which has been studied for other variants of bin packing [21,5]. We specifically show the following.…”

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confidence: 99%

Improved Approximation for Vector Bin Packing

Bansal¹,

Eliáš²,

Khan³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the d-dimensional vector bin packing problem, a well-studied generalization of bin packing arising in resource allocation and scheduling problems. Here we are given a set of d-dimensional vectors v1, . . . , vn in [0,1] d , and the goal is to pack them into the least number of bins so that for each bin B, the sum of the vectors in it is at most 1 in every dimension, i.e., || v i ∈B vi||∞ ≤ 1. For the 2-dimensional case we give an asymptotic approximation guarantee of 1 + ln(1.5) + ǫ ≈ (1.405 + ǫ), improving upon the previous bound of 1 + ln 2 + ǫ ≈ (1.693 + ǫ). We also give an almost tight (1.5 + ǫ) absolute approximation guarantee, improving upon the previous bound of 2 [23]. For the d-dimensional case, we get a 1.5 + ln() + ǫ ≈ 0.807 + ln(d + 1) + ǫ guarantee, improving upon the previous (1+ln d+ǫ) guarantee [2]. Here (1 +ln d) was a natural barrier as rounding-based algorithms can not achieve better than d approximation. We get around this by exploiting various structural properties of (near)-optimal packings, and using multi-objective multi-budget matching based techniques and expanding the Round & Approx framework to go beyond rounding-based algorithms. Along the way we also prove several results that could be of independent interest.

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Rectangle packing with one-dimensional resource augmentation

Cited by 42 publications

References 20 publications

Two for One: Tight Approximation of 2D Bin Packing

Two for One: Tight Approximation of 2D Bin Packing

A (5/3 + ε)-Approximation for Strip Packing

Improved Approximation for Vector Bin Packing

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