Title: The knapsack and its applications Author: Romana Linkeová Department: Department of Algebra Supervisor: doc. Mgr. Pavel Příhoda, Ph.D., Department of Algebra Abstract: This thesis is focused on various aspects of cryptosystems based on NP (non-deterministic polynomial) complete knapsack problem. From the theory of complexity point of view, the less known parts of the proof of knapsack problem NP completeness are shown in detail. From the cryptographical point of view, a demonstration of breaking of the Merkle-Hellman cryptosystem (the basic de- sign of knapsack-type cryptosystems) is provided, showing that poor parameters choice can lead to easy obtaining of the whole private key. Another contribution of this thesis consists in a presented proposal of a new cryptosystem concept based on the matrix 0-1 knapsack problem. This concept was developed in order to prevent known attacks, however, in the thesis we provide a proof analogous to J. C. Lagarias and A. M. Odlyzko, 1985, which shows that an attack based on the LLL algorithm will be successful on the majority of the matrix 0-1 kna- psack problem cryptosystems. Finally, a list of modern cryptosystems based on the knapsack problem is provided and a cryptanalysis thereof is given. Keywords: knapsack problem, NP complete problems, LLL algorithm 1

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programming (IP) has been and continues to be widely used by industries to minimize cost and effectively manage resources. Faster computers and innovative IP techniques have enabled the solution to many large-scale IPs. However, IPs are NP-hard and many IPs require exponential time to solve. Lifting is one of the most widely used techniques that helps to reduce computational time and is widely applied in today's commercial IP software. Lifting was first introduced by Gomory for bounded integer programs and a theoretical and computationally intractible technique to simultaneously lift sets of variables was introduced by Zemel in 1978. This thesis presents a new algorithm called the Maximal Simultaneous Lifting Algorithm (MSLA), to simultaneously uplift sets of binary integer variables into a cover inequality. These lifted inequalities result in strong inequalities that are facet defining under fairly moderate assumptions. A computational study shows that this algorithm can find numerous strong inequalities for random Knapsack (KP) instances. The pre-processing time observed for these instances is less than 1/50th of a second, which is negligible. These simultaneously lifted inequalities are easy to find and incorporating these cuts to KP instances reduced the solution time by an average of 41%. Therefore, implementing MSLA should be highly beneficial for large real-world problems.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd W. Easton / Integer programming is an important discipline in operation research that positively impacts society. Unfortunately, no algorithm currently exists to solve IP's in polynomial time. Researchers are constantly developing new techniques, such as cutting planes, to help solve IPs faster. For example, DeLissa discovered the existence of equality cuts limited to zero and one coefficients for the multiple knapsack equality problem (MKEP). An equality cut is an improper cut because every feasible point satisfies the equality. However, such a cut always reduces the dimension of the linear relaxation space by at least one. This thesis introduces lifted equality cuts, which can have coefficients greater than or equal to two. Two main theorems provide the conditions for the existence of lifted equalities. These theorems provide the foundation for The Algorithm of Lifted Equality Cuts (ALEC), which finds lifted equality cuts in quadratic time. The computational study verifies the benefit of lifted equality cuts in random MKEP instances. ALEC generated millions of lifted equality cuts and reduced the solution time by an average of 15%. To the best of the author's knowledge, ALEC is the first algorithm that has found over 30.7 million cuts on a single problem, while reducing the solving time by 18%.

The thesis describes the principles of dynamic programming and it's application to concrete problems. (The travelling salesman problem, the knapsack problem, the shortest path priblem,the set covering problem.)

The diploma thesis focuses on tracing in layout by handlers between assembly lines in new plant for corporation Continental Automotive Czech Republic ltd, where boosters are produced. The theoretical part involves definitions of logistics, supply chain, material flow and handling equipment. Furthermore, methods of mathematic programming and software equipment are described, such as quadratic assignment problem, knapsack problem, travelling salesman problem from graph theory. In the practical part the situation in corporation has been analyzed and the data prepared for further examination. Then layout of plant and internal processes are evaluated and an appropriate model or concept of solution is selected. Subsequently, application in MS Excel is created with support of VBA scripts (3 kinds of layouts). The user manipulates with application followed by Solver for implementation of a new solution into practice. Finally, the models are interpreted and verified by Lingo. The focus of the thesis is the design of a layout change of a new plant including the description of tracing.

In which ways can the process of trimming dispersion coated board products be optimized so as to reduce material waste and increase production efficiency? This is the question that this master thesis report seeks to answer. In paper production, alot of waste is generated when cutting production reels into customer reels. Some material waste are necessary in order to ensure good quality, however a large amount of the wastecould be reduced if the cutting process was to be optimized. During this project, carried out at a forest company, a mathematical optimization model was developed in order to reduce waste and save costs. This model is based on a cutting stock problem using column generation approach. It provides as its output cutting patterns and an optimal allocation of rolls for production purposes, which implies minimizing the number production rolls needed.The visualization of the results could also be used to achieve optimal stock levels, and easier keep track on how to use the material available in stock. Findings show that there are potential savings to be done. Simulations suggest an implementation of this model could result in material savings of around 7 %. This could also translateto environmental savings in CO2, where every decrease of one tonne material equals to adecrease in CO2emissions of 500 kg

No description available.

Engine-triggered tasks are real-time tasks that are released when the crankshaft arrives at certain positions in its path of rotation. This makes the rate of release of these jobs a function of the crankshaft's angular speed and acceleration. In addition, several properties of the engine triggered tasks like the execution time and deadlines are dependent on the speed profile of the crankshaft. Such tasks are referred to as adaptive-variable rate (AVR) tasks. Existing methods to calculate the worst-case demand of AVR tasks are either inaccurate or computationally intractable. We propose a method to efficiently calculate the worst-case demand of AVR tasks by transforming the problem into a variant of the knapsack problem. We then propose a framework to systematically narrow down the search space associated with finding the worst-case demand of AVR tasks. Experimental results show that our approach is at least 10 times faster, with an average runtime improvement of 146 times for randomly generated task sets when compared to the state-of-the-art technique. / Master of Science / Real-time systems require temporal correctness along with accuracy. This notion of temporal correctness is achieved by specifying deadlines to each of the tasks. In order to ensure that all the deadlines are met, it is important to know the processor requirement, also known as demand, of a task over a given interval. For some tasks, the demand is not constant, instead it depends on several external factors. For such tasks, it becomes necessary to calculate the worst-case demand. Engine-triggered tasks are activated when the crankshaft in an engine is at certain points in its path of rotation. This makes their activation rate dependent on the angular speed and acceleration of the crankshaft. In addition, several properties of the engine triggered tasks like the execution time and deadlines are dependent on the speed profile of the crankshaft. Such tasks are referred to as adaptive-variable rate (AVR) tasks. Existing methods to calculate the worst-case demand of AVR tasks are either inaccurate or computationally intractable. We propose a method to efficiently calculate the worst-case demand of AVR tasks by transforming the problem into a variant of the knapsack problem. We then propose a framework to systematically narrow down the search space associated with finding the worst-case demand of AVR tasks. Experimental results show that our approach is at least 10 times faster, with an average runtime improvement of 146 times for randomly generated task sets when compared to the state-of-the-art technique.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IPs) are mathematical models that can provide an optimal solution to a variety of different problems. They have been used to reduce costs and optimize organizations. Additionally, IPs are NP-complete resulting in many IPs that cannot be solved. Cutting planes or valid inequalities have been used to decrease the time required to solve IPs. Lifting is a technique that strengthens existing valid inequalities. Lifting inequalities can result in facet defining inequalities, which are the theoretically strongest valid inequalities. Because of these properties, lifting procedures are used in software to reduce the time required to solve an IP. The thesis introduces a new algorithm for exact synchronized simultaneous uplifting over an arbitrary initial inequality for knapsack problems. Synchronized Simultaneous Lifting (SSL) is a pseudopolynomial time algorithm requiring O(nb+n[superscript]3) effort to solve. It exactly uplifts two sets simultaneously into an initial arbitrary valid inequality and creates multiple inequalities of a particular form. This previously undiscovered class of inequalities generated by SSL can be facet defining. A small computational study shows that SSL is quick to execute, requiring on average less than a quarter of a second. Additionally, applying SSL inequalities to a knapsack problem enabled commercial software to solve problems that it could not solve without them.

Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick's Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick's Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick's Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research.