The target of this thesis is to explain the Travelling Salesman Problem and also create a special program, which will be able to make calculations using the heuristics GENIUS. The Travelling Salesman Problem will be described from two different points of view. The first one is the historical description of the idea of the Travelling Salesman Problem and later will be the problem will be described with some of the very wide number of the calculation methods. For the explanation of the methods, in the thesis there has been chosen some of the algorithms which belong to that methods. The heuristics and also the exact algorithms will be explained. The focus of this thesis is on the heuristics called GENIUS and also in the creation of the program which can calculate it. The program works first with the GENI algorithm and after that it works with US post-optimization algorithm. The program will be described from the point of view of the user and the manual will be written as well. The program will be tested on two different examples and will be compared with the exact algorithm.

Praktische Optimierungsprobleme beinhalten oft mehrere gleichberechtigte, sich jedoch widersprechende Kriterien. Beispielsweise will man bei einer Reise zugleich möglichst schnell ankommen, sie soll aber auch nicht zu teuer sein. Im ersten Teil dieser Arbeit wird die algorithmische Beherrschbarkeit solcher mehrkriterieller Optimierungsprobleme behandelt. Es werden zunächst verschiedene Lösungsbegriffe diskutiert und auf ihre Schwierigkeit hin verglichen. Interessanterweise stellt sich heraus, dass diese Begriffe für ein einkriterielles Problem stets gleich schwer sind, sie sich ab zwei Kriterien allerdings stark unterscheiden könen (außer es gilt P = NP). In diesem Zusammenhang wird auch die Beziehung zwischen Such- und Entscheidungsproblemen im Allgemeinen untersucht. Schließlich werden neue und verbesserte Approximationsalgorithmen für verschieden Varianten des Problems des Handlungsreisenden gefunden. Dabei wird mit Mitteln der Diskrepanztheorie eine Technik entwickelt, die ein grundlegendes Hindernis der Mehrkriteriellen Optimierung aus dem Weg schafft: Gegebene Lösungen so zu kombinieren, dass die neue Lösung in allen Kriterien möglichst ausgewogen ist und gleichzeitig die Struktur der Lösungen nicht zu stark zerstört wird. Der zweite Teil der Arbeit widmet sich verschiedenen Aspekten von Gleichungssystemen für (formale) Sprachen. Einerseits werden konjunktive und Boolesche Grammatiken untersucht. Diese sind Erweiterungen der kontextfreien Grammatiken um explizite Durchschnitts- und Komplementoperationen. Es wird unter anderem gezeigt, dass man bei konjunktiven Grammatiken die Vereinigungsoperation stark einschränken kann, ohne dabei die erzeugte Sprache zu ändern. Außerdem werden bestimmte Schaltkreise untersucht, deren Gatter keine Wahrheitswerte sondern Mengen von Zahlen berechnen. Für diese Schaltkreise wird das Äquivalenzproblem betrachtet, also die Frage ob zwei gegebene Schaltkreise die gleiche Menge berechnen oder nicht. Es stellt sich heraus, dass, abhängig von den erlaubten Gattertypen, die Komplexität des Äquivalenzproblems stark variiert und für verschiedene Komplexitätsklassen vollständig ist, also als (parametrisierter) Vertreter für diese Klassen stehen kann. / Practical optimization problems often comprise several incomparable and conflicting objectives. When booking a trip using several means of transport, for instance, it should be fast and at the same time not too expensive. The first part of this thesis is concerned with the algorithmic solvability of such multiobjective optimization problems. Several solution notions are discussed and compared with respect to their difficulty. Interestingly, these solution notions are always equally difficulty for a single-objective problem and they differ considerably already for two objectives (unless P = NP). In this context, the difference between search and decision problems is also investigated in general. Furthermore, new and improved approximation algorithms for several variants of the traveling salesperson problem are presented. Using tools from discrepancy theory, a general technique is developed that helps to avoid an obstacle that is often hindering in multiobjective approximation: The problem of combining two solutions such that the new solution is balanced in all objectives and also mostly retains the structure of the original solutions. The second part of this thesis is dedicated to several aspects of systems of equations for (formal) languages. Firstly, conjunctive and Boolean grammars are studied, which are extensions of context-free grammars by explicit intersection and complementation operations, respectively. Among other results, it is shown that one can considerably restrict the union operation on conjunctive grammars without changing the generated language. Secondly, certain circuits are investigated whose gates do not compute Boolean values but sets of natural numbers. For these circuits, the equivalence problem is studied, i.\,e.\ the problem of deciding whether two given circuits compute the same set or not. It is shown that, depending on the allowed types of gates, this problem is complete for several different complexity classes and can thus be seen as a parametrized) representative for all those classes.

The first part of this thesis deals with the approximability of the traveling salesman problem. This problem is defined on a complete graph with edge weights, and the task is to find a Hamiltonian cycle of minimum weight that visits each vertex exactly once. We study the most important multiobjective variants of this problem. In the multiobjective case, the edge weights are vectors of natural numbers with one component for each objective, and since weight vectors are typically incomparable, the optimal Hamiltonian cycle does not exist. Instead we consider the Pareto set, which consists of those Hamiltonian cycles that are not dominated by some other, strictly better Hamiltonian cycles. The central goal in multiobjective optimization and in the first part of this thesis in particular is the approximation of such Pareto sets. We first develop improved approximation algorithms for the two-objective metric traveling salesman problem on multigraphs and for related Hamiltonian path problems that are inspired by the single-objective Christofides' heuristic. We further show arguments indicating that our algorithms are difficult to improve. Furthermore we consider multiobjective maximization versions of the traveling salesman problem, where the task is to find Hamiltonian cycles with high weight in each objective. We generalize single-objective techniques to the multiobjective case, where we first compute a cycle cover with high weight and then remove an edge with low weight in each cycle. Since weight vectors are often incomparable, the choice of the edges of low weight is non-trivial. We develop a general lemma that solves this problem and enables us to generalize the single-objective maximization algorithms to the multiobjective case. We obtain improved, randomized approximation algorithms for the multiobjective maximization variants of the traveling salesman problem. We conclude the first part by developing deterministic algorithms for these problems. The second part of this thesis deals with redundancy properties of complete sets. We call a set autoreducible if for every input instance x we can efficiently compute some y that is different from x but that has the same membership to the set. If the set can be split into two equivalent parts, then it is called weakly mitotic, and if the splitting is obtained by an efficiently decidable separator set, then it is called mitotic. For different reducibility notions and complexity classes, we analyze how redundant its complete sets are. Previous research in this field concentrates on polynomial-time computable reducibility notions. The main contribution of this part of the thesis is a systematic study of the redundancy properties of complete sets for typical complexity classes and reducibility notions that are computable in logarithmic space. We use different techniques to show autoreducibility and mitoticity that depend on the size of the complexity class and the strength of the reducibility notion considered. For small complexity classes such as NL and P we use self-reducible, complete sets to show that all complete sets are autoreducible. For large complexity classes such as PSPACE and EXP we apply diagonalization methods to show that all complete sets are even mitotic. For intermediate complexity classes such as NP and the remaining levels of the polynomial-time hierarchy we establish autoreducibility of complete sets by locally checking computational transcripts. In many cases we can show autoreducibility of complete sets, while mitoticity is not known to hold. We conclude the second part by showing that in some cases, autoreducibility of complete sets at least implies weak mitoticity. / Der erste Teil dieser Arbeit widmet sich der Approximierbarkeit des Traveling Salesman Problems, bei welchem man in vollständigen Graphen mit Kantengewichten eine Rundreise mit minimalem Gewicht sucht. Es werden die wichtigsten mehrkriteriellen Varianten dieses Problems betrachtet, bei denen die Kantengewichte aus Vektoren natürlicher Zahlen mit einer Komponente pro Kriterium bestehen. Verschiedene Rundreisen sind bei mehrkriteriellen Kantengewichten häufig unvergleichbar, und dementsprechend existiert oft keine eindeutige optimale Rundreise. Stattdessen fasst man jene Rundreisen, zu denen jeweils keine eindeutig bessere Rundreise existiert, zu der sogenannten Pareto-Menge zusammen. Die Approximation solcher Pareto-Mengen ist die zentrale Aufgabe in der mehrkriteriellen Optimierung und in diesem Teil der Arbeit. Durch Techniken, die sich an Christofides' Heuristik aus der einkriteriellen Approximation orientieren, werden zunächst verbesserte Approximationsalgorithmen für das zweikriterielle metrische Traveling Salesman Problem auf Multigraphen und für analog definierte Hamiltonpfadprobleme entwickelt. Außerdem werden Argumente gegen eine signifikante Verbesserung dieser Algorithmen aufgezeigt. Weiterhin werden mehrkriterielle Maximierungsvarianten des Traveling Salesman Problems betrachtet, bei denen man Rundreisen mit möglichst großem Gewicht in jedem Kriterium sucht. Es werden einkriterielle Techniken auf den mehrkriteriellen Fall übertragen, bei denen man zunächst eine Kreisüberdeckung mit hohem Gewicht berechnet und anschließend pro Kreis die Kante mit dem niedrigsten Gewicht löscht. Die Auswahl einer solchen Kante pro Kreis ist im mehrkriteriellen Fall nicht trivial, weil mehrkriterielle Gewichtsvektoren häufig unvergleichbar sind. Es wird ein allgemeines Lemma entwickelt, welches dieses Problem löst und damit eine Übertragung der einkriteriellen Maximierungsalgorithmen auf den mehrkriteriellen Fall ermöglicht. Dadurch ergeben sich verbesserte, randomisierte Approximationsalgorithmen für die mehrkriteriellen Maximierungsvarianten des Traveling Salesman Problems. Abschließend werden zu diesen Problemvarianten deterministische Algorithmen entwickelt. Der zweite Teil dieser Arbeit widmet sich Redundanzeigenschaften vollständiger Mengen. Eine Menge heißt autoreduzierbar, wenn zu jeder Instanz x eine von x verschiedene Instanz y mit der gleichen Zugehörigkeit zu der Menge effizient berechnet werden kann. Ist die Menge in zwei äquivalente Teile aufspaltbar, so heißt sie schwach mitotisch, und ist diese Aufspaltung durch einen effizient entscheidbaren Separator erreichbar, so heißt sie mitotisch. Zu verschiedenen Reduktionen und Komplexitätsklassen wird die Frage betrachtet, wie redundant ihre vollständigen Mengen sind. Während sich vorherige Forschung in diesem Gebiet hauptsächlich auf Polynomialzeitreduktionen konzentriert, liefert diese Arbeit eine systematische Analyse der Redundanzeigenschaften vollständiger Mengen für typische Komplexitätsklassen und solche Reduktionen, die sich in logarithmischem Raum berechnen lassen. Je nach Größe der Komplexitätsklasse und Stärke der Reduktion werden dabei verschiedene Techniken eingesetzt. Für kleine Komplexitätsklassen wie beispielsweise NL und P werden selbstreduzierbare, vollständige Mengen benutzt, um Autoreduzierbarkeit aller vollständigen Mengen nachzuweisen, während für große Komplexitätsklassen wie beispielsweise PSPACE und EXP Diagonalisierungsmethoden sogar die Mitotizität vollständiger Mengen zeigen. Für dazwischen liegende Komplexitätsklassen wie beispielsweise NP und die übrigen Level der Polynomialzeithierarchie wird Autoreduzierbarkeit vollständiger Mengen über lokales Testen von Berechnungstranskripten gezeigt. Während in vielen Fällen Autoreduzierbarkeit vollständiger Mengen gezeigt werden kann, bleibt häufig die Frage offen, ob diese Mengen auch mitotisch sind. Abschließend wird gezeigt, dass in einigen Fällen Autoreduzierbarkeit vollständiger Mengen zumindest schwache Mitotizität impliziert.

The Ant System is a new meta-heuristic method particularly appropriate to solve hard combinatorial optimization problems. It is a population-based, nature-inspired approach exploiting positive feedback as well as local information and has been applied successfully to a variety of combinatorial optimization problem classes. The Ant System consists of a set of cooperating agents (artificial ants) and a set of rules that determine the generation, update and usage of local and global information in order to find good solutions. As the structure of the Ant System highly suggests a parallel implementation of the algorithm, in this paper two parallelization strategies for an Ant System implementation are developed and evaluated: the synchronous parallel algorithm and the partially asynchronous parallel algorithm. Using the Traveling Salesman Problem a discrete event simulation is performed, and both strategies are evaluated on the criteria "speedup", "efficiency" and "efficacy". Finally further improvements for an advanced parallel implementation are discussed. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

The Tour Construction Framework (TCF) integrates both global and local heuristics in a complementary framework in order to efficiently solve the Travelling Salesman Problem (TSP). Most tour construction heuristics are strictly local in nature. However, the experimental method presented in this research includes a global heuristic to efficiently solve the TSP. The Global Path (GP) component and Super Node (SN) component comprise the TCF. Each component heuristic is tuned with one or more parameters. Genetic Algorithms (GA) are used to train the collection of parameters for the TCF components on subsets of benchmark TSPs. The GA results are used to run the TCF on the full TSP instances. The performance of the TCF is evaluated for speed, accuracy, and computational complexity, and it is compared against six mainstream TSP solvers: Lin-Kernighan-Helsgaun (LKH-2), 2-Opt, Greedy, Boruvka, Quick-Boruvka, and Nearest Neighbor. The empirical study demonstrates the effectiveness of the TCF in achieving near-optimal solutions for the TSP with reasonable costs.

No / This paper presents a new class of heuristics which embed an exact algorithm within the framework of a local search heuristic. This approach was inspired by related heuristics which we developed for a practical problem arising in electronics manufacture. The basic idea of this heuristic is to break the original problem into small subproblems having similar properties to the original problem. These subproblems are then solved using time intensive heuristic approaches or exact algorithms and the solution is re-embedded into the original problem. The electronics manufacturing problem where we originally used the embedded local search approach, contains the Travelling Salesman Problem (TSP) as a major subproblem. In this paper we further develop our embedded search heuristic, HyperOpt, and investigate its performance for the TSP in comparison to other local search based approaches. We introduce an interesting hybrid of HyperOpt and 3-opt for asymmetric TSPs which proves more efficient than HyperOpt or 3-opt alone. Since pure local search seldom yields solutions of high quality we also investigate the performance of the approaches in an iterated local search framework. We examine iterated approaches of Large-Step Markov Chain and Variable Neighbourhood Search type and investigate their performance when used in combination with HyperOpt. We report extensive computational results to investigate the performance of our heuristic approaches for asymmetric and Euclidean Travelling Salesman Problems. While for the symmetric TSP our approaches yield solutions of comparable quality to 2-opt heuristic, the hybrid methods proposed for asymmetric problems seem capable of compensating for the time intensive embedded heuristic by finding tours of better average quality than iterated 3-opt in many less iterations and providing the best heuristic solutions known, for some instance classes.

Minimal length of a travelling salesman's problem had been studied in this diploma these. Travelling salesman must come trough each place just once and then go back to the starting place. This problem can be illustrated as a problem of graph theory, such that places are the vertices, roads are the edges, distances of roads are the lengths of edges. The optimal travelling salesman's problem tour is the shortest Hamiltionian cycle in the graph. It is a classical NP-complete problem. There is no algorithm that solves this problem in polynomial time. This problem can be solved by using various approximation algorithms, they offer less time consumption and lowest quality than optimization. This diploma these had been dedicated to approximation algorithms, for example: nearest neighbor method, minimal spanning tree method, Christofide's method, 2-opt., genetic algorithm, etc.

A very wide-ranging subgroup of vehicle routing problems from the graph theory is a common and frequent problem handled daily by transport companies, airline businesses, hi-tech companies with planning drilling of printed circuits boards or other companies from different industries. During numerous previous researches of these problems a lot of analyses were made and many solutions proposed -- of which an outline is in this paper. Some of them giving better or worse results in longer or shorter computing time. In spite of the fact that the processors and new technologies performance is increasing, with some algorithms we cannon compute the result in a reasonable time. That is why this paper is asking a question, if there can be found a fitting algorithm which could be applied on different and faster processing unit structures so it could be ensured a multiple computing speed increase so far. The analysis was carried out using computer experiments on a new build and implemented branch and bound algorithm with a matrix rate reduction.

Constructing the genomic median of several given genomes is crucial in developing evolutionary trees, since the genomic median provides an estimate for the ordering of the genes in a common ancestor of the given genomes. This is due to the fact that the content of DNA molecules is often similar, but the difference is mainly in the order in which the genes appear in various genomes. The mutations that affect this ordering are called genome rearrangements, and many structural differences between genomes can be studied using genome rearrangements. In this thesis our main focus is on applying combinatorial optimization techniques to genomic median problems, with particular emphasis on the breakpoint distance as a measure of the difference between two genomes. We will study different variations of the breakpoint median problem from signed to unsigned, unichromosomal to multichromosomal, and linear to circular to mixed. We show how these median problems can be formulated in terms of problems in combinatorial optimization, and take advantage of well-known combinatorial optimization techniques and apply these powerful methods to study various median problems. Some of these median problems are polynomial and many are NP-hard. We find efficient algorithms and approximation methods for median problems based on well-known combinatorial optimization structures. The focus is on algorithmic and combinatorial aspects of genomic medians, and how they can be utilized to obtain optimal median solutions.

Constructing the genomic median of several given genomes is crucial in developing evolutionary trees, since the genomic median provides an estimate for the ordering of the genes in a common ancestor of the given genomes. This is due to the fact that the content of DNA molecules is often similar, but the difference is mainly in the order in which the genes appear in various genomes. The mutations that affect this ordering are called genome rearrangements, and many structural differences between genomes can be studied using genome rearrangements. In this thesis our main focus is on applying combinatorial optimization techniques to genomic median problems, with particular emphasis on the breakpoint distance as a measure of the difference between two genomes. We will study different variations of the breakpoint median problem from signed to unsigned, unichromosomal to multichromosomal, and linear to circular to mixed. We show how these median problems can be formulated in terms of problems in combinatorial optimization, and take advantage of well-known combinatorial optimization techniques and apply these powerful methods to study various median problems. Some of these median problems are polynomial and many are NP-hard. We find efficient algorithms and approximation methods for median problems based on well-known combinatorial optimization structures. The focus is on algorithmic and combinatorial aspects of genomic medians, and how they can be utilized to obtain optimal median solutions.