Šiame darbe nagrinėjama negiljotininio, dvimačio, stačiakampių pjaustymo uždavinio atliekų minimizavimo problema ir jos sprendimo metodai. Dėl uždavinio kombinatorinio sudėtingumo neįmanoma tiksliai ir visais atvejais pateikti optimalų jo sprendinį, todėl pasirinkti apytiksliai sprendimo metodai. Uždavinys sprendžiamas metaeuristiniais hibridiniais genetiniu ir modeliuojamo atkaitinimo algoritmais apjungtais su euristiniais „Žemiausio kairėn užpildymo“ ir „Žemiausio tarpo“, kuris yra originali „Geriausiai tinkamo“ metodo modifikacija. Taip pat realizuojami minėti euristiniai algoritmai atskirai nuo hibridinių. Atliekama šių metodų lyginamoji analizė bei jų parametrų ir pradinių sąlygų parinkimo įtakos tyrimas sprendinio kokybei. Suformuojama ir pateikiama metodika pjaustymo uždavinių sprendimui. / A non – guillotinable, two – dimensional, rectangular cutting stock problem is being introduced in this paper and its solving methods either. Due to the combinatorial complexity of a problem, it is impossible to solve it optimally for every instance. Consequently an aproximate methods have been chosen. The problem is solved by metaheuristic genetic and simulated annealing methods hybridised with heuristic „Bottom Left Fill“ and „Lowest Gap“, which is an originally modified version of „Best Fit“ algorithm. The same heuristic algorithms are implemented separately from hybridised ones. A comparation analysis of these methods is done and the influence on solution quality depending on the selection of algorithms parameters and its initial conditions is considered. The methodology of solving cutting stock problems is being formulated and presented.

Let there be m parallel machines and n jobs to be scheduled non-preemptively. A job j scheduled on machine i takes p_{i,j} time units to complete, where 1 ≤ i ≤ m and 1 ≤ j ≤ n. For a given schedule, the makespan is the completion time of a machine that finishes last. The goal is to produce a schedule of all n jobs with minimum makespan. This is known as the makespan problem on unrelated parallel machines (UPMs), denoted as R||C_{max}. In this thesis, we focus on subclasses of R||C_{max}. Our research consists of two components. First, a survey of theoretic results for R||C_{max} with a focus on approximation algorithms is presented. Second, we present exact polynomial-time algorithms and approximation algorithms for some subclasses of R||C_{max}. For instance, we present k-approximation algorithms on par with or better than the best known for certain subclasses of R||C_{max}.

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability.

In this thesis, we show results for some well-studied problems from learning theory and combinatorial optimization. Learning Parities under the Uniform Distribution: We study the learnability of parities in the agnostic learning framework of Haussler and Kearns et al. We show that under the uniform distribution, agnostically learning parities reduces to learning parities with random classification noise, commonly referred to as the noisy parity problem. Together with the parity learning algorithm of Blum et al, this gives the first nontrivial algorithm for agnostic learning of parities. We use similar techniques to reduce learning of two other fundamental concept classes under the uniform distribution to learning of noisy parities. Namely, we show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables and learning of k-juntas reduces to learning noisy parities of k variables. Agnostic Learning of Halfspaces: We give an essentially optimal hardness result for agnostic learning of halfspaces over rationals. We show that for any constant ε finding a halfspace that agrees with an unknown function on 1/2+ε fraction of examples is NP-hard even when there exists a halfspace that agrees with the unknown function on 1-ε fraction of examples. This significantly improves on a number of previous hardness results for this problem. We extend the result to ε = 2[superscript-Ω(sqrt{log n})] assuming NP is not contained in DTIME(2[superscript(log n)O(1)]). Majorities of Halfspaces: We show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem. This also implies a hardness result for learning halfspaces with a high rate of adversarial noise even if the learning algorithm can output any efficiently computable hypothesis. Max-Clique, Chromatic Number and Min-3Lin-Deletion: We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph (also referred to as the Max-Clique problem) and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2[superscript(log n)3/4+γ] in an n vertex graph, assuming NP is not contained in BPTIME(2[superscript(log n)O(1)]). This improves the hardness factor of n/2[superscript (log n)1-γ'] for some small (unspecified) constant γ' > 0 shown by Khot. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2[superscript sqrt{log n}] for this problem, improving upon the hardness factor of (log n)[superscriptβ] shown by Hastad, for some small (unspecified) constant β > 0. The hardness results for Max-Clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given by Khot. Monotone Multilinear Boolean Circuits for Bipartite Perfect Matching: A monotone Boolean circuit is said to be multilinear if for any AND gate in the circuit, the minimal representation of the two input functions to the gate do not have any variable in common. We show that monotone multilinear Boolean circuits for computing bipartite perfect matching require exponential size. In fact we prove a stronger result by characterizing the structure of the smallest monotone multilinear Boolean circuits for the problem.

Even though the necessity of water quality monitoring systems is increasing, and though mobile watery quality monitoring systems using the combination of automatic measuring devices and autonomous vehicles is becoming available, research on effective deployment of such systems is not studied well. The locations or paths to take the measurement are one of the most important design factors to maximize the performance of water quality monitoring systems, and they needs to be optimized to maximize the monitoring performance. To solve these optimization problems, multi-objective genetic algorithms were proposed and developed. The proposed optimization procedures were applied to hypothetical circular lakes and Lake Pontchartrain in order to obtain optimal monitoring locations, straight monitoring paths, and higher-order monitoring paths under various conditions. Also, the effect of various parameters such as the speed of a monitoring vessel, the weights of possible scenarios, and etc. are investigated. The optimization models found optimal solutions efficiently while reflecting various effects of complex physical settings. The results from the optimizations show that distribution of possible source locations is an important factor that affects optimal solutions greatly. In a closed water body, wind is major forcing that determines hydrodynamics and contaminant transport, and it affects optimal solutions as well. Straight monitoring lines do not perform very well due to their incapability to cover the irregular boundaries of water bodies. Higher-order optimal monitoring paths overcome this difficulty and perform well up to a comparable level of a few stationary monitoring locations even under realistic and transient conditions.

With recent and continuing advances in bioinformatics, the volume of sequence data has increased tremendously. Along with this increase, there is a growing need to develop efficient algorithms to process such data in order to make useful and important discoveries. Careful analysis of genomic data will benefit science and society in numerous ways, including the understanding of protein sequence functions, early detection of diseases, and finding evolutionary relationships that exist among various organisms. Most sequence analysis problems arising from computational genomics and evolutionary biology fall into the class of NP-complete problems. Advances in exact and approximate algorithms to address these problems are critical. In this thesis, we investigate a novel graph theoretical model that deals with fundamental evolutionary problems. The model allows incorporation of the evolutionary operations ``insertion', ``deletion', and ``substitution', and various parameters such as relative distances and weights. By varying appropriate parameters and weights within the model, several important combinatorial problems can be represented, including the weighted supersequence, weighted superstring, and weighted longest common sequence problems. Consequently, our model provides a general computational framework for solving a wide variety of important and difficult biological sequencing problems, including the multiple sequence alignment problem, and the problem of finding an evolutionary ancestor of multiple sequences. In this thesis, we develop large scale combinatorial optimization techniques to solve our graph theoretical model. In particular, we formulate the problem as two distinct but related models: constrained network flow problem and weighted node packing problem. The integer programming models are solved in a branch and bound setting using simultaneous column and row generation. The methodology developed will also be useful to solve large scale integer programming problems arising in other areas such as transportation and logistics.

In this thesis, we use integer programming techniques to solve the resource constrained shortest path problem (RCSPP) which seeks a minimum cost path between two nodes in a directed graph subject to a finite set of resource constraints. Although NP-hard, the RCSPP is extremely useful in practice and often appears as a subproblem in many decomposition schemes for difficult optimization problems. We begin with a study of the RCSPP polytope for the single resource case and obtain several new valid inequality classes. Separation routines are provided, along with a polynomial time algorithm for constructing an auxiliary conflict graph which can be used to separate well known valid inequalities for the node packing polytope. We establish some facet defining conditions when the underlying graph is acyclic and develop a polynomial time sequential lifting algorithm which can be used to strengthen one of the inequality classes. Next, we outline a branch-and-cut algorithm for the RCSPP. We present preprocessing techniques and branching schemes which lead to strengthened linear programming relaxations and balanced search trees, and the majority of the new inequality classes are generalized to consider multiple resources. We describe a primal heuristic scheme that uses fractional solutions, along with the current incumbent, to search for new feasible solutions throughout the branch-and-bound tree. A computational study is conducted to evaluate several implementation choices, and the results demonstrate that our algorithm outperforms the default branch-and-cut algorithm of a leading integer programming software package. Finally, we consider the dial-a-flight problem (DAFP), a new vehicle routing problem that arises in the context of on-demand air transportation and is concerned with the scheduling of a set of travel requests for a single day of operations. The DAFP can be formulated as an integer multicommodity network flow model consisting of several RCSPPs linked together by set partitioning constraints which guarantee that all travel requests are satisfied. Therefore, we extend our branch-and-cut algorithm for the RCSPP to solve the DAFP. Computational experiments with practical instances provided by the DayJet Corporation verify that the extended algorithm also outperforms the default branch-and-cut algorithm of a leading integer programming software package.

Random graph processes are most often used to investigate theoretical questions about random graphs. A randomised algorithm can be defined specifically for the purpose of finding some structure in a graph, such as a matching, a colouring or a particular kind of sub graph. Properties of the related random graph process then suggest properties, or bounds on properties, of the structure. In this thesis, we use a random graph process to analyse a particular load balancing algorithm from theoretical computer science. By doing so, we demonstrate that random graph processes may also be used to analyse other algorithms and systems of a random nature, from areas such as computer science, telecommunications and other areas of engineering and mathematics. Moreover, this approach can lead to theoretical results on the performance of algorithms that are difficult to obtain by other methods. In the course of our analysis we are also led to some results of the first kind, relating to the structure of the random graph. / The particular algorithm that we analyse is a randomised algorithm for an off-line load balancing problem with two choices. The load balancing algorithm, in an initial stage, mirrors an algorithm which finds the k-core of a graph. This latter algorithm and the related random graph process have been previously analysed by Pittel, Spencer and Wormald, using a differential equation method, to determine the thresholds for the existence of a k-core in a random graph. We modify their approach by using a random pseudograph model due to Bollobas and Frieze, and Chvatal, in place of the uniform random graph. This makes the analysis somewhat simpler, and leads to a shortened derivation of the thresholds and other properties of k-cores.(For complete abstract open document)

Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Computer Science, 2009. / Includes bibliographical references.

A teoria dos grafos é uma ferramenta matemática muito utilizada na resolução de problemas algorítmicos e computacionais em que se quer modelar conjuntos de elementos e relações entre estes elementos. Sistemas naturais e tecnológicos de diversos domínios podem ser representados matematicamente por grafos que possuem propriedades estatísticas bem conhecidas, sendo uma destas propriedades a distribuição de graus dos vértices do grafo seguindo a lei de potência (power law). Exemplos destes grafos, conhecidos como grafos power law são a internet, World-Wide Web, as redes sociais, redes biológicas. No contexto de problemas algorítmicos em grafos, estamos interessados em problemas computacionalmente difíceis de serem resolvidos que pertencem à classe NP-Difícil (ou NP-Hard), mais especificamente no problema de cobertura por vértices. Neste trabalho será estudado experimentalmente o comportamento de um algoritmo baseado em uma estratégia gulosa para o problema de cobertura de vértices e compararemos com outro algoritmo de aproximação e com a solução exponencial ótima. Em particular esta solução será aplicada e analisada em redes complexas. / Graph theory is a mathematical tool used in solving many algorithmic and computational problems in that both sets of model elements and relationships between these elements. Most natural and technological systems can be mathematically modeled by graph having many well known properties, in particular the power law distribution of the vertex degree sequence. Examples of such graphs, called power law graphs are the Internet, World-Wide Web, social networks, biological networks. In the context of algorithmic problems on graphs, we are interested in problems in class NP-Hard, more specifically in the vertex cover problem. This work will be studied experimentally the behavior of an algorithm based on a greedy strategy for the vertex cover problem and compare with other approximation algorithms and with the exponential optimal solution. In particular this solution will be applied and analyzed in complex networks.