Approximating Smallest Containers for Packing Three-dimensional Convex Objects

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimal volume containers for the objects described

Improved Pseudo-Polynomial-Time Approximation for Strip Packing

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the "empty space" left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well

Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins

We explore approximation algorithms for the d-dimensional geometric bin packing problem (dBP). Caprara [Caprara, 2008] gave a harmonic-based algorithm for dBP having an asymptotic approximation ratio (AAR) of (T_?)^{d-1} (where T_? ? 1.691). However, their algorithm doesn\u27t allow items to be rotated. This is in contrast to some common applications of dBP, like packing boxes into shipping containers. We give approximation algorithms for dBP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm having AAR T_?^d. We next give a more sophisticated harmonic-based algorithm, which we call HGaP_k, having AAR (T_?)^{d-1}(1+?). This gives an AAR of roughly 2.860 + ? for 3BP with rotations, which improves upon the best-known AAR of 4.5. In addition, we study the multiple-choice bin packing problem that generalizes the rotational case. Here we are given n sets of d-dimensional cuboidal items and we have to choose exactly one item from each set and then pack the chosen items. Our algorithms also work for the multiple-choice bin packing problem. We also give fast and simple approximation algorithms for the multiple-choice versions of dD strip packing and dD geometric knapsack

Algoritmos de aproximação para problemas de alocação de instalações e outros problemas de cadeia de fornecimento

Orientadores: Flávio Keidi Miyazawa, Maxim SviridenkoTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O resumo poderá ser visualizado no texto completo da tese digitalAbstract: The abstract is available with the full electronic documentDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

Algorithmes d'approximation pour des programmes linéaires et les problèmes de Packing avec des contraintes géometriques

In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is a model from convex optimization. This problem includes a large class of linear programs. The algorithms generate approximately feasible solutions within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and O(M epsilon{-2} ln (M epsilon^{-1}))$iterations, respectively, where in each iteration a block problem which depends on the specific application has to be solved. Both algorithms, applied to linear programs, can result in column generation algorithms. In Chapter 3, we implement an algorithm for the so-called max-min-resource sharing problem. This is a certain convex optimization problem which, similar to the problem in Chapter 1, includes a large class of linear programs. The implementation, which is included in the appendix, is done in C++. We use the implementation in the context of an AFPTAS for Strip Packing in order to evaluate dynamic optimization of a parameter in the algorithm, namely the step length used for interpolation. We compare our choice to the static step length proposed in the analysis of the algorithm and conclude that dynamic optimization of the step length significantly reduces the number of iterations. In Chapter 4, we study two closely related scheduling problems, namely non-preemptive scheduling with fixed jobs and scheduling with non-availability for sequential jobs on m identical machines under the makespan objective, where m is constant. For the first problem, which does not admit an FPTAS unless P=NP, we obtain a new PTAS. For the second problem, we show that a suitable restriction (namely the permanent availability of one machine) is necessary to obtain a bounded approximation ratio. For this restriction, which does not admit an FPTAS unless P=NP, we present a PTAS; we also discuss the complexity of various special cases. In total, the results are basically best possible. In Chapter 5, we continue the studies from Chapter 4 where now the number m of machines is part of the input, which makes the problem algorithmically harder. Scheduling with fixed jobs does not admit an approximation ratio better than 3/2, unless P=NP; here we obtain an approximation ratio of 3/2+epsilon for any epsilon>0. For scheduling with non-availability, we require a constant percentage of the machines to be permanently available. This restriction also does not admit an approximation ratio better than 3/2 unless P=NP; we also obtain an approximation ratio of$3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible. Finally, in Chapter 6, we conclude with some remarks and open research problems

Algorithmes d'approximation pour des programmes linéaires et les problèmes de Packing avec des contraintes géometriques

In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is a model from convex optimization. This problem includes a large class of linear programs. The algorithms generate approximately feasible solutions within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and O(M epsilon{-2} ln (M epsilon^{-1}))$iterations, respectively, where in each iteration a block problem which depends on the specific application has to be solved. Both algorithms, applied to linear programs, can result in column generation algorithms. In Chapter 3, we implement an algorithm for the so-called max-min-resource sharing problem. This is a certain convex optimization problem which, similar to the problem in Chapter 1, includes a large class of linear programs. The implementation, which is included in the appendix, is done in C++. We use the implementation in the context of an AFPTAS for Strip Packing in order to evaluate dynamic optimization of a parameter in the algorithm, namely the step length used for interpolation. We compare our choice to the static step length proposed in the analysis of the algorithm and conclude that dynamic optimization of the step length significantly reduces the number of iterations. In Chapter 4, we study two closely related scheduling problems, namely non-preemptive scheduling with fixed jobs and scheduling with non-availability for sequential jobs on m identical machines under the makespan objective, where m is constant. For the first problem, which does not admit an FPTAS unless P=NP, we obtain a new PTAS. For the second problem, we show that a suitable restriction (namely the permanent availability of one machine) is necessary to obtain a bounded approximation ratio. For this restriction, which does not admit an FPTAS unless P=NP, we present a PTAS; we also discuss the complexity of various special cases. In total, the results are basically best possible. In Chapter 5, we continue the studies from Chapter 4 where now the number m of machines is part of the input, which makes the problem algorithmically harder. Scheduling with fixed jobs does not admit an approximation ratio better than 3/2, unless P=NP; here we obtain an approximation ratio of 3/2+epsilon for any epsilon>0. For scheduling with non-availability, we require a constant percentage of the machines to be permanently available. This restriction also does not admit an approximation ratio better than 3/2 unless P=NP; we also obtain an approximation ratio of$3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible. Finally, in Chapter 6, we conclude with some remarks and open research problems

On Discrete Hyperbox Packing

Bin packing is a very important and popular research area in the computer science field. Past work showed many good and real-world packing algorithms. How- ever, due to the complexity of the problem in multiple-dimensional bin packing, also called hyperbox packing, we need more practical packing algorithms for its real-world applications. In this dissertation, we extend 1D packing algorithms to hyperbox packing prob- lems via a general framework that takes two inputs of a 1D packing algorithm and an instance of hyperbox packing problem and outputs a hyperbox packing algorithm. The extension framework significantly enriches the family of hyperbox-packing algorithms, generates many framework-based algorithms, and simultaneously calls for the analysis for those algorithms. We also analyze the performance of a couple of framework-based algorithms from two perspectives of worst-case performance and average-case performance. In worst- case analysis, we use the worst-case performance ratio as our metric and analyze the relationship of the ratio of framework-based algorithms and that of the corresponding 1D algorithms. We also compare their worst-case performance against two baselines: strip optimal algorithms and optimal algorithms. In average-case analysis, we use expected waste as a metric, analyze the waste of optimal hyperbox packing algorithms, and estimate the asymptotic forms of the waste for framework-based algorithms

Studies in Efficient Discrete Algorithms

This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity