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}.

A graph is said to be 2-edge-connected if it remains connected after the deletion of any single edge. Given an unweighted bridgeless graph G with n vertices, the minimum size 2-edge-connected spanning subgraph problem (2EC) is that of finding a 2-edge-connected spanning subgraph of G with the minimum number of edges. This problem has important applications in the design of survivable networks. However, because the problem is NP-hard, it is unlikely that efficient methods exist for solving it. Thus efficient methods that find solutions that are provably close to optimal are sought. In this thesis, an approximation algorithm is presented for 2EC on bridgeless cubic graphs which guarantees to be within 5/4 of the optimal solution value, improving on the previous best proven approximation guarantee of 5/4+ε for this problem. We also focus on the linear programming (LP) relaxation of 2EC, which provides important lower bounds for 2EC in useful solution techniques like branch and bound. The “goodness” of this lower bound is measured by the integrality gap of the LP relaxation for 2EC, denoted by α2EC. Through a computational study, we find the exact value of α2EC for graphs with small n. Moreover, a significant improvement is found for the lower bound on the value of α2EC for bridgeless subcubic graphs, which improves the known best lower bound on α2EC from 9/8 to 8/7.

In many emerging applications, data streams are monitored in a network environment. Due to limited communication bandwidth and other resource constraints, a critical and practical demand is to online compress data streams continuously with quality guarantee. Although many data compression and digital signal processing methods have been developed to reduce data volume, their super-linear time and more-than-constant space complexity prevents them from being applied directly on data streams, particularly over resource-constrained sensor networks. In this thesis, we tackle the problem of online quality guaranteed compression of data streams using fast linear approximation (i.e., using line segments to approximate a time series). Technically, we address two versions of the problem which explore quality guarantees in different forms. We develop online algorithms with linear time complexity and constant cost in space. Our algorithms are optimal in the sense that they generate the minimum number of segments that approximate a time series with the required quality guarantee. To meet the resource constraints in sensor networks, we also develop a fast algorithm which creates connecting segments with very simple computation. The low cost nature of our methods leads to a unique edge on the applications of massive and high speed streaming environment, low bandwidth networks, and heavily constrained nodes in computational power (e.g., tiny sensor nodes). We implement and evaluate our methods in the application of an acoustic wireless sensor network.

In this dissertation, we examine specific network connectivity problems, and achieve improved approximation algorithm and integrality gap results for them. We introduce an important new, highly useful and applicable, network connectivity problem - the Vital Core Connectivity Problem (VCC). Despite its many practical uses, this problem has not been previously studied. We present the first constant factor approximation algorithm for VCC, and provide an upper bound on the integrality gap of its linear programming relaxation. We also introduce a new, useful, extension of the minimum spanning tree problem, called the Extended Minimum Spanning Tree Problem (EMST), that is based on a special case of VCC; and provide both a polynomial-time algorithm and a complete linear description for it. Furthermore, we show how to generalize this new problem to handle numerous disjoint vital cores, providing the first complete linear description of, and polynomial-time algorithm for, the generalized problem. We examine the Survivable Network Design Problem (SNDP) with multiple copies of edges allowed in the solution (multi-SNDP), and present a new approximation algorithm for which the approximation guarantee is better than that of the current best known for certain cases of multi-SNDP. With our method, we also obtain improved bounds on the integrality gap of the linear programming relaxation of the problem. Furthermore, we show the application of these results to variations of SNDP. We investigate cases where the optimal values of multi-SNDP and SNDP are equal; and we present an improvement on the previously best known integrality gap bound and approximation guarantee for the special case of SNDP with metric costs and low vertex connectivity requirements, as well as for the similar special case of the Vertex Connected Survivable Network Design Problem (VC-SNDP). The quality of the results that one can obtain for a given network design problem often depends on its integer linear programming formulation, and, in particular, on its linear programming relaxation. In this connection, we investigate formulations for the Steiner Tree Problem (ST). We propose two new formulations for ST, and investigate their strength in terms of their associated integrality gaps.

Στην εργασία αυτή ασχολούμαστε με θέματα κοινωνικής επιλογής και πιο συγκεκριμένα με συμβιβαστικές ψηφοφορίες στις οποίες κάθε ψηφοφόρος ψηφίζει ένα (πιθανόν κενό) σύνολο υποψηφίων και το αποτέλεσμα είναι ένα σύνολο υποψηφίων πλήθους k, για δεδομένο k (π.χ. εκλογή επιτροπής). Εξετάζουμε τον κανόνα minimax σε συμβιβαστικές ψηφοφορίες, στις οποίες το αποτέλεσμα αντιπροσωπεύει ένα συμβιβασμό μεταξύ των προτιμήσεων των ψηφοφόρων, με την έννοια ότι η μέγιστη απόσταση μεταξύ των προτιμήσεων οποιουδήποτε ψηφοφόρου και του αποτελέσματος είναι όσο το δυνατό μικρότερη. Αυτός ο κανόνας έχει δύο μειονεκτήματα. Πρώτον, ο υπολογισμός του αποτελέσματος που ελαχιστοποιεί τη μέγιστη απόσταση από κάθε ψηφοφόρο είναι ένα υπολογιστικά δύσκολο πρόβλημα και δεύτερον, οποιοσδήποτε αλγόριθμος που πάντα επιστρέφει ένα τέτοιο αποτέλεσμα, δίνει στους ψηφοφόρους κίνητρο να πουν ψέματα για την πραγματική τους προτίμηση, με σκοπό να βελτιώσουν την απόσταση τους από το τελικό αποτέλεσμα. Για να ξεπεράσουμε αυτά τα μειονεκτήματα χρησιμοποιούμε προσεγγιστικούς αλγορίθμους, δηλαδή αλγορίθμους που παράγουν αποτέλεσμα που αποδεδειγμένα προσεγγίζει την minimax απόσταση για κάθε δοσμένο στιγμιότυπο. Τέτοιοι αλγόριθμοι μπορούν να χρησιμοποιηθούν σαν εναλλακτικοί κανόνες ψηφοφορίας. Παρουσιάζουμε ένα 2-προσεγγιστικό αλγόριθμο πολυωνυμικού χρόνου, ο οποίος υπολογίζει το αποτέλεσμα στρογγυλοποιώντας ντετερμινιστικά τη λύση του χαλαρωμένου γραμμικού προγράμματος μέσω του οποίου εκφράζουμε το πρόβλημά μας. Ο καλύτερος προηγούμενος προσεγγιστικός αλγόριθμος επιτύγχανε λόγο απόδοσης 3 και συνεπώς το παραπάνω αποτέλεσμα αποτελεί σημαντική βελτίωση. Επιπλέον ασχολούμαστε με προσεγγιστικούς αλγορίθμους που είναι ανθεκτικοί σε χειραγώγηση είτε από μεμονωμένους ψηφοφόρους είτε από ομάδες ψηφοφόρων. Τέτοιοι αλγόριθμοι δεν προσφέρουν κίνητρο στους ψηφοφόρους να δηλώσουν ψευδώς τις προτιμήσεις τους με σκοπό να βελτιώσουν την απόστασή τους από το τελικό αποτέλεσμα. Μια τέτοια μελέτη εντάσσεται στα πλαίσια της έρευνας που γίνεται τα τελευταία χρόνια πάνω στο σχεδιασμό προσεγγιστικών αλγοριθμικών μηχανισμών χωρίς χρήματα. Συμπληρώνουμε προηγούμενα αποτελέσματα με νέα πάνω και κάτω φράγματα για strategyproof και group-strategyproof αλγορίθμους. / We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some fixed k, e.g., committee elections. We are interested in the minimax approval voting rule in which the outcome represents a compromise among the preferences of the voters, in the sense that the maximum distance between the preference of any voter and the outcome is as small as possible. This voting rule has two main drawbacks. First, computing an outcome that minimizes the maximum distance is computationally hard. Furthermore, any algorithm that always returns such an outcome provides incentives to voters to misreport their true preferences. In order to circumvent these drawbacks, we consider approximation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given instance. Such algorithms can be considered as alternative voting rules. We present a polynomial-time 2-approximation algorithm that uses a natural linear programming relaxation for the underlying optimization problem and deterministically rounds the fractional solution in order to compute the outcome; this result improves upon the previously best known algorithm that has an approximation ratio of 3. We are furthermore interested in approximation algorithms that are resistant to manipulation by (coalitions of) voters, i.e., algorithms that do not motivate voters to misreport their true preferences in order to improve their distance from the outcome. This study falls within the recently initiated line of research on approximate mechanism design without money. We complement previous results in the literature with new upper and lower bounds on strategyproof and group-strategyproof algorithms.

[pt] Nesta dissertação é apresentado um framework para desenvolver critérios de split para lidar com atributos nominais multi-valorados em árvores de decisão. Critérios gerados por este framework podem ser implementados para rodar em tempo polinomial no número de classes e valores, com garantia teórica de produzir um split próximo do ótimo. Apresenta-se também um estudo experimental, utilizando datasets reais, onde o tempo de execução e acurácia de métodos oriundos do framework são avaliados. / [en] In this dissertation we propose a framework for designing splitting criteria for handling multi-valued nominal attributes for decision trees. Criteria derived from our framework can be implemented to run in polynomial time in the number of classes and values, with theoretical guarantee of producing a split that is close to the optimal one. We also present an experimental study, using real datasets, where the running time and accuracy of the methods obtained from the framework are evaluated.

Les travaux de cette thèse concernent la résolution d'un problème de planification dans un réseau de distribution à deux échelons intégrant la gestion de stocks de produits périssables, le dimensionnement de lots, des alternatives d'approvisionnement. La livraison s'effectue directement entre un fournisseur et son client, sans tournée avec une flotte homogène de véhicules. Nous proposons un programme linéaire mixte, une heuristique constructive (déterministe) et une heuristique réactive randomisée. Pour certaines instances, le solveur de programme linéaire mixte ne fournit pas une bonne solution réalisable dans la limite de temps définie ou prend beaucoup de temps. Les heuristiques proposées sont rapides mais ne donnent pas de bonnes solutions pour certaines instances. Pour améliorer la qualité des solutions des heuristiques, la descente à voisinage variable (VND), la recherche locale itérative (ILS) et la recherche locale itérative à démarrages multiples (MS-ILS) sont développées.Toutes ces méthodes ont été incluses dans un APS (Advanced Planning System) et sont comparées avec CPLEX sur des instances extraites de bases de données réelles. Un générateur aléatoire d'instances est conçu pour plus de diversité pour les tests. Une relaxation lagrangienne est implémentée pour comparer les solutions des instances, pour lesquelles CPLEX ne fournit pas une bonne solution réalisable dans le temps imparti, avec les autres méthodes. Une heuristique lagrangienne, utilisant la relaxation lagrangienne et une heuristique de réparation, est également développée / This work presents a planning problem in a distribution network incorporating two levels inventory management of perishable products, lot-sizing, multi-sourcing and transport capacity with a homogeneous fleet of vehicles. A mixed integer linear programming (MILP) a greedy heuristic and a reactive randomized heuristic are developed to solve this real planning problem. There are some instances for which the solver CPLEX cannot give a good upper bound within the limited time and for other instances it takes a lot of time to solve MILP. The heuristics are alternatives to the mixed integer linear program to quickly solve some large instances taking into account original and difficult constraints. For some instances the gap between the solutions of the solver (MILP) and the heuristics becomes quite significant. The variable neighborhood descent (VND), the iterated local search (ILS) and the multi-start iterated local search (MS-ILS) are implemented. These methods are included in an APS (Advanced Planning System) and compared with a MILP solver. The instances are derived from actual data or built using a random generator of instances to have wider diversity for computational evaluation. A lagrangian relaxation is developed to compare the solutions of the instances, for which CPLEX cannot give a good upper bound within the limited time, with the other methods (greedy heuristic, VND, ILS and MS-ILS). A lagrangian heuristic is proposed; the solution of lagrangian relaxation is used to build a feasible solution with a repair heuristic

Este trabalho tem por objetivo estudar os problemas computacionais que surgem ao se relacionar grafos com superfícies bidimensionais, dando especial atenção aos problemas do número de cruzamentos mínimo no plano (CROSSING NUMBER) e a problemas relacionados ao desenho de grafos em livros. Apresentamos uma redução do problema MULTICUT para CROSSING NUMBER, além de um resultado de complexidade em grafos de comparabilidade baseado em um resultado conhecido para desenhos em livros. / The objective of this text is to study computational problems that emerge from the relation between graphs and bidimensional surfaces, giving special attention to the crossing number problem and graph drawings on books. We present a reduction from MULTICUT to CROSSING NUMBER, in addition to a complexity result on comparability graphs based on a known result about drawings on books.

Orientador: Eduardo Candido Xavier / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-24T00:15:42Z (GMT). No. of bitstreams: 1 Silveira_JeffersonLuizMoisesda_D.pdf: 2236708 bytes, checksum: 8e569408c2f068347058e36031689c3a (MD5) Previous issue date: 2013 / Resumo: Neste trabalho estamos interessados em problemas de empacotamento e roteamento. Assumindo a hipótese de que P ? NP, sabemos que não existem algoritmos eficientes para resolver tais problemas. Além de algoritmos exatos, duas das abordagens para resolver tais problemas são Algoritmos Aproximados e Heurísticas. Nesta tese mostramos algoritmos baseados nestas três abordagens para ambos os problemas, de empacotamento e roteamento. Os dois primeiros problemas atacados foram generalizações de problemas clássicos de empacotamento: O problema da mochila bidimensional e o problema de empacotamento em faixas. Estes foram generalizados adicionando restrições na forma de carregamento e descarregamento dos itens no recipiente (restrições estas, que aparecem no contexto de problemas de roteamento). O terceiro problema é uma combinação de problemas de empacotamento e roteamento. Neste caso, atacamos uma generalização do clássico Pickup and Delivery Problem. Propomos os primeiros resultados de aproximação para algumas versões dos problemas de empacotamento supracitados. Além disto, apresentamos algumas abordagens práticas para o terceiro problema. As heurísticas foram avaliadas através de experimentos computacionais comparando os seus resultados com algoritmos exatos / Abstract: In this work we are interested in packing and routing problems. Assuming P ? NP, we have that there are no efficient algorithms to deal with such problems. Besides exact algorithms, two approaches to solve such problems are Approximation Algorithms and Heuristics. In this thesis we show algorithms using these three approaches for both packing and routing problems. The first two addressed problems are generalizations of classical packing problems: The Two Dimensional Knapsack problem and the Strip Packing problem. These problems were generalized by adding constraints on the way the items can be inserted/removed into/from the bin (These constraints appear in the context of routing problems). The third problem is combination of packing and routing problems. It is a generalization of the classical Pickup and Delivery problem. We propose the first approximation results for some packing problems. Besides that, we present some practical algorithms for the third problem. The heuristics were assessed through computational experiments by comparing their results with exact algorithms / Doutorado / Ciência da Computação / Doutor em Ciência da Computação

The operational scheduling at coal handling facilities is of the utmost importance to ensure that the coal consuming processes are supplied with a constant feed of good quality coal. Although the Sasol Coal Handling Facility (CHF) were not designed to perform coal blending during the coal handling process, CHF has to blend the different sources to ensure that the quality of the feed supplied is of a stable nature. As a result, the operation of the plant has become an extremely complex process. Consequently, human intelligence is no longer sufficient to perform coal handling scheduling and therefore a scheduling model is required to ensure optimal plant operation and optimal downstream process performance. After various attempts to solve the scheduling model optimally, i.e. with exact solution methods, it was found that it is not possible to accurately model the complexities of CHF in such a way that the currently available exact solvers can solve it in an acceptable operational time. Various alternative solution approaches are compared, in terms of solution quality and execution speed, using a simplified version of the CHF scheduling problem. This investigation indicates that the Simulated Annealing (SA) metaheuristic is the most efficient solution method to provide approximate solutions. The metaheuristic solution approach allows one to model the typical sequential thoughts of a control room operator and sequential operating procedures. Thus far, these sequential rules could not be modelled in the simultaneous equation environment required for exact solution methods. An SA metaheuristic is developed to solve the practical scheduling model. A novel SA approach is applied where, instead of the actual solution being used for neighbourhood solution representation, the neighbours are indirectly represented by the rules used to generate neighbourhood solutions. It is also found that the initial temperature should not be a fixed value, but should be a multiple of the objective function value of the initial solution. An inverse arctan-based cooling schedule function outperforms traditional cooling schedules as it provides the required diversification and intensification behaviour of the SA. The scheduling model solves within 45 seconds and provides good, practically executable results. The metaheuristic approach to scheduling is therefore successful as the plant complexities and intricate operational philosophies can be accurately modelled using the sequential nature of programming languages and provides good approximate optimal solutions in a short solution time. Tests done with live CHF data indicate that the metaheuristic solution outperforms the current scheduling methodologies applied in the business. The implementation of the scheduler will lead to a more stable factory feed, which will increase production yields and therefore increase company profits. By reducing the amount of coal re-handling (in terms of throw-outs and load-backs at mine bunkers), the scheduler will reduce the coal handling facility’s annual operating cost by approximately R4.6 million (ZAR). Furthermore, the approaches discussed in this document can be applied to any continuous product scheduling environment. Additional information available on a CD stored at Level 3 of the Merensky Library. / Dissertation (MEng (Industrial Engineering))--University of Pretoria, 2011. / Industrial and Systems Engineering / unrestricted