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

Advances and Novel Approaches in Discrete Optimization

Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms

The Computational Complexity of Linear Optics

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.Comment: 94 pages, 4 figure

Hexarray: A Novel Self-Reconfigurable Hardware System

Evolvable hardware (EHW) is a powerful autonomous system for adapting and finding solutions within a changing environment. EHW consists of two main components: a reconfigurable hardware core and an evolutionary algorithm. The majority of prior research focuses on improving either the reconfigurable hardware or the evolutionary algorithm in place, but not both. Thus, current implementations suffer from being application oriented and having slow reconfiguration times, low efficiencies, and less routing flexibility. In this work, a novel evolvable hardware platform is proposed that combines a novel reconfigurable hardware core and a novel evolutionary algorithm. The proposed reconfigurable hardware core is a systolic array, which is called HexArray. HexArray was constructed using processing elements with a redesigned architecture, called HexCells, which provide routing flexibility and support for hybrid reconfiguration schemes. The improved evolutionary algorithm is a genome-aware genetic algorithm (GAGA) that accelerates evolution. Guided by a fitness function the GAGA utilizes context-aware genetic operators to evolve solutions. The operators are genome-aware constrained (GAC) selection, genome-aware mutation (GAM), and genome-aware crossover (GAX). The GAC selection operator improves parallelism and reduces the redundant evaluations. The GAM operator restricts the mutation to the part of the genome that affects the selected output. The GAX operator cascades, interleaves, or parallel-recombines genomes at the cell level to generate better genomes. These operators improve evolution while not limiting the algorithm from exploring all areas of a solution space. The system was implemented on a SoC that includes a programmable logic (i.e., field-programmable gate array) to realize the HexArray and a processing system to execute the GAGA. A computationally intensive application that evolves adaptive filters for image processing was chosen as a case study and used to conduct a set of experiments to prove the developed system robustness. Through an iterative process using the genetic operators and a fitness function, the EHW system configures and adapts itself to evolve fitter solutions. In a relatively short time (e.g., seconds), HexArray is able to evolve autonomously to the desired filter. By exploiting the routing flexibility in the HexArray architecture, the EHW has a simple yet effective mechanism to detect and tolerate faulty cells, which improves system reliability. Finally, a mechanism that accelerates the evolution process by hiding the reconfiguration time in an “evolve-while-reconfigure” process is presented. In this process, the GAGA utilizes the array routing flexibility to bypass cells that are being configured and evaluates several genomes in parallel