New Approximability Results for Two-Dimensional Bin Packing

We study the two-dimensional bin packing problem: Given a list of n rectangles the objective is to find a feasible, i.e. axis-parallel and non-overlapping, packing of all rectangles into the minimum number of unit sized squares, also called bins. Our problem consists of two versions; in the first version it is not allowed to rotate the rectangles while in the other it is allowed to rotate the rectangles by 90∘, i.e. to exchange the widths and the heights. Two-dimensional bin packing is a generalization of its one-dimensional counterpart and is therefore strongly NP-hard. Furthermore Bansal et al. showed that even an APTAS is ruled out for this problem, unless P=NP. This lower bound of asymptotic approximability was improved by Chlebik and Chlebikova to values 1+1/3792 and 1+1/2196 for the version with and without rotations, respectively. On the positive side there is an asymptotic 1.69.. approximation by Caprara without rotations and an asymptotic 1.52... approximation by Bansal et al.for both versions. We give a new asymptotic upper bound for both versions of our problem: For any fixed ε and any instance that fits optimally into OPT bins, our algorithm computes a packing into (3/2+ε)⋅OPT+69 bins in the version without rotations and (3/2+ε)⋅OPT+39 bins in the version with rotations. The algorithm has polynomial running time in the input length. In our new technique we consider an optimal packing of the rectangles into the bins. We cut a small vertical or horizontal strip out of each bin and move the intersecting rectangles into additional bins. This enables us to either round the widths of all wide rectangles, or the heights of all long rectangles in this bin. After this step we round the other unrounded side of these rectangles and we achieve a solution with a simple structure and only few types of rectangles. Our algorithm initially rounds the instance and computes a solution that nearly matches the modified optimal solution

Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA\u2714]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS\u2705] obtained an APTAS for this problem. Let ? be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by ? opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ? ? ? 1.692. Bansal and Khan [SODA\u2714] conjectured that ? = 4/3. The conjecture, if true, will imply a (4/3+?)-approximation algorithm for 2BP. According to convention, for a given constant ? > 0, a rectangle is large if both its height and width are at least ?, and otherwise it is called skewed. We make progress towards the conjecture by showing ? = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on ? was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS

A tale of two packing problems : improved algorithms and tighter bounds for online bin packing and the geometric knapsack problem

In this thesis, we deal with two packing problems: the online bin packing and the geometric knapsack problem. In online bin packing, the aim is to pack a given number of items of different size into a minimal number of containers. The items need to be packed one by one without knowing future items. For online bin packing in one dimension, we present a new family of algorithms that constitutes the first improvement over the previously best algorithm in almost 15 years. While the algorithmic ideas are intuitive, an elaborate analysis is required to prove its competitive ratio. We also give a lower bound for the competitive ratio of this family of algorithms. For online bin packing in higher dimensions, we discuss lower bounds for the competitive ratio and show that the ideas from the one-dimensional case cannot be easily transferred to obtain better two-dimensional algorithms. In the geometric knapsack problem, one aims to pack a maximum weight subset of given rectangles into one square container. For this problem, we consider online approximation algorithms. For geometric knapsack with square items, we improve the running time of the best known PTAS and obtain an EPTAS. This shows that large running times caused by some standard techniques for geometric packing problems are not always necessary and can be improved. Finally, we show how to use resource augmentation to compute optimal solutions in EPTAS-time, thereby improving upon the known PTAS for this case.In dieser Arbeit betrachten wir zwei Packungsprobleme: Online Bin Packing und das geometrische Rucksackproblem. Bei Online Bin Packing versucht man, eine gegebene Menge an Objekten verschiedener Größe in die kleinstmögliche Anzahl an Behältern zu packen. Die Objekte müssen eins nach dem anderen gepackt werden, ohne zukünftige Objekte zu kennen. Für eindimensionales Online Bin Packing beschreiben wir einen neuen Algorithmus, der die erste Verbesserung gegenüber dem bisher besten Algorithmus seit fast 15 Jahren darstellt. Während die algorithmischen Ideen intuitiv sind, ist eine ausgefeilte Analyse notwendig um das Kompetitivitätsverhältnis zu beweisen. Für Online Bin Packing in mehreren Dimensionen geben wir untere Schranken für das Kompetitivitätsverhältnis an und zeigen, dass die Ideen aus dem eindimensionalen Fall nicht direkt zu einer Verbesserung führen. Beim geometrischen Rucksackproblem ist es das Ziel, eine größtmögliche Teilmenge gegebener Rechtecke in einen einzelnen quadratischen Behälter zu packen. Für dieses Problem betrachten wir Approximationsalgorithmen. Für das Problem mit quadratischen Objekten verbessern wir die Laufzeit des bekannten PTAS zu einem EPTAS. Die langen Laufzeiten vieler Standardtechniken für geometrische Probleme können also vermieden werden. Schließlich zeigen wir, wie Ressourcenvergrößerung genutzt werden kann, um eine optimale Lösung in EPTAS-Zeit zu berechnen, was das bisherige PTAS verbessert.Google PhD Fellowshi

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one