In the past few years, there has been a strong and growing interest in evaluating the expected behavior of what we call combinatorial bin packing problems. A combinatorial bin packing problem consists of a number of items of various sizes and value ratios (value per unit of size) along with a collection of bins of fixed capacity into which the items are to be packed. The packing must be done in such a way that the sum of the sizes of the items into a given bin does not exceed the capacity of that bin. Moreover, an item must either be packed into a bin in its entirety or not at all: this "all or nothing" requirement is why these problems are characterized as being combinatorial. The objective of the packing is to optimize a given criterion Junction. Here optimize means either maximize or minimize, depending on the problem. We study two problems that fit into this framework: the Knapsack Problem and the Minimum Sum of Squares Problem. Both of these problems are known to be in the class of NP-hard problems and there is ample reason to suspect that these problems do not admit of efficient exact solution. We obtain results concerning the performance of heuristics under the assumption that the inputs are random samples from some distribution. For the Knapsack Problem, we develop four heuristics, two of which are on-line and two off-line. All four heuristics are shown to be asymptotically optimal in expectation when the item sizes and value ratios are assumed to be independent and uniform. One heuristic is shown to be asymptotically optimal in expectation when the item sizes are uniformly distributed and the value ratios are exponentially distributed. The amount of time required by these heuristics is no more than proportional to the amount of time required to sort the items in order of nonincreasing value ratios. For the Minimum Sum of Squares Problem, we develop two heuristics, both of which are off-line. Both of these heuristics are shown to be asymptotically optimal in expectation when the sizes of the items input are assumed uniformly distributed.

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Thesis (M.S.)--Rochester Institute of Technology, 1997. / Typescript. Includes bibliographical references (leaves 43-44).

Thesis (doctoral)--Universität Breslau.

Thesis (M.S.)--Rochester Institute of Technology, 1994. / Typescript. Bibliography: leaves 33-35.

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The bin packing problem has been the corner stone of approximation algorithms and has been extensively studied starting from the early seventies. In the classical bin packing problem, we are given a list of real numbers in the range (0, 1], the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. In this thesis we study approximation algorithms for three generalizations of bin packing: geometric bin packing, vector bin packing and weighted bipartite edge coloring. In two-dimensional (2-D) geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. Geometric packing has vast applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. We consider the widely studied orthogonal packing case, where the items must be placed in the bin such that their sides are parallel to the sides of the bin. Here two variants are usually studied, (i) where the items cannot be rotated, and (ii) they can be rotated by 90 degrees. We give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for the versions with and without rotations. We have also shown the limitations of rounding based algorithms, ubiquitous in bin packing algorithms. We have shown that any algorithm that rounds at least one side of each large item to some number in a constant size collection values chosen independent of the problem instance, cannot achieve an asymptotic approximation ratio better than 3/2. In d-dimensional vector bin packing (VBP), each item is a d-dimensional vector that needs to be packed into unit vector bins. The problem is of great significance in resource constrained scheduling and also appears in recent virtual machine placement in cloud computing. Even in two dimensions, it has novel applications in layout design, logistics, loading and scheduling problems. We obtain a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for 2-D VBP. We also obtain a polynomial time algorithm with almost tight (absolute) approximation ratio of $1+\ln(1.5)$ for 2-D VBP. For $d$ dimensions, we give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(d/2) + 1.5 \approx \ln d+0.81$. We also consider vector bin packing under resource augmentation. We give a polynomial time algorithm that packs vectors into $(1+\epsilon)Opt$ bins when we allow augmentation in (d - 1) dimensions and $Opt$ is the minimum number of bins needed to pack the vectors into (1,1) bins. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite graph $G=(V,E)$ with weights $w: E \rightarrow [0,1]$. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. This problem is motivated by rearrangeability of 3-stage Clos networks which is very useful in various applications in interconnected networks and routing. We show a polynomial time approximation algorithm that returns a proper weighted coloring with at most $\lceil 2.2223m \rceil$ colors where $m$ is the minimum number of unit sized bins needed to pack the weight of all edges incident at any vertex. We also show that if all edge weights are $>1/4$ then $\lceil 2.2m \rceil$ colors are sufficient.

Šio darbo tikslas – sukurti trimačių dėžių su papildomu orientacijos erdvėje apribojimu pakavimo optimizavimo algoritmą, o realizavus jį programinėmis priemonėmis – ištirti jo efektyvumą su atsitiktinai sugeneruotų kraunamų dėžių rinkiniais ir prie skirtingų algoritmo veikimo parametrų. Taip pat sukurti ir pakavimo optimizavimo sprendinio vizualizavimo trimatėje erdvėje programinę įrangą. / Presented work covers one of the most complex areas of combinatorial optimization – three dimensional bin packing problem. Solution methods of this problem are applied in the real world from logistics, packing optimization to VLSI circuit and automobile engineering. Several heuristic packing algorithms suggested by other authors are analyzed. Approach based on tree-search and wall building strategy is chosen to create a 3D packing optimization algorithm. A bin orientation in space restriction is added to classical 3D bin packing problem to make it more complex and more suited for real world applications. A prototype of created algorithm is created and tested with randomly generated data collections. Each data sample is processed with and without bin orientation in space restriction. Influence of restriction and maximal tree width on packing efficiency and computational time is statistically analyzed. Visualization tool based on Microsoft Direct X technology is created to view results of packing optimization.

An online algorithm must make decisions immediately and irrevocably based only on a part of the input without any knowledge of the future part of the input. We introduce the competitive analysis of online algorithms, a standard worst-case analysis, and present main results of this analysis on the problem of online Bin Packing and on some of its variants. In Bin Packing, a sequence of items of size up to 1 arrives to be packed into the minimal number of unit capacity bins. Mainly, we focus on Colored Bin Packing in which items have also a color and we cannot pack two items of the same color adjacently in a bin. For Colored Bin Packing, we improve some previous results on the problem with two colors and present the first results for arbitrarily many colors. Most notably, in the important case when all items have size zero, we give an optimal 1.5-competitive algorithm. For items of arbitrary size we present a lower bound of 2.5 and a 3.5-competitive algorithm. Powered by TCPDF (www.tcpdf.org)