In this paper, we concentrate on computing several critical budgets for interdiction of the multicommodity network flows, and studying the interdiction effects of the changes on budget. More specifically, we first propose general interdiction models of the multicommodity flow problem, with consideration of both node and arc removals and decrease of their capacities. Then, to perform the vulnerability analysis of networks, we define the function F(R) as the minimum amount of unsatisfied demands in the resulted network after worst-case interdiction with budget R. Specifically, we study the properties of function F(R), and find the critical budget values, such as , the largest value under which all demands can still be satisfied in the resulted network even under the worst-case interdiction, and , the least value under which the worst-case interdiction can make none of the demands be satisfied. We prove that the critical budget for completely destroying the network is not related to arc or node capacities, and supply or demand amounts, but it is related to the network topology, the sets of source and destination nodes, and interdiction costs on each node and arc. We also observe that the critical budget is related to all of these parameters of the network. Additionally, we present formulations to estimate both and . For the effects of budget increasing, we present the conditions under which there would be extra capabilities to interdict more arcs or nodes with increased budget, and also under which the increased budget has no effects for the interdictor. To verify these results and conclusions, numerical experiments on 12 networks with different numbers of commodities are performed.

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints.

Minimum Concave Cost Multicommodity Network Design Problem arises in many application areas, such as transportation planning, distributed energy system and especially both circuit and packet switching backbone network design. Exact concave optimization algorithms have been developed, but these methods are applicable if the network size is small. Therefore, these problems are usually solved by non-exact iterative methods. In this thesis work, methods proposed for circuit switching and packet switching network design are evaluated in detail. After a comprehensive literate survey, Yaged&rsquo / s Linearization, Minoux greedy and Minoux accelerated greedy methods are found to be applicable to circuit switching network design when both solution quality and computational time is considered. Previously, it has been found that Minoux greedy methods may create routings with cycles and in order to eliminate these cycles a modification has been proposed. In this work, this modification is extended and evaluated in detail. Similarly, Gerla and Kleinrock&rsquo / s Concave Branch Elimination, Gersht&rsquo / s greedy and Stacey&rsquo / s Concave Link Elimination methods are investigated within the context of packet switching network design. All of these methods consider aggregate flows on each link simultaneously re-routing more than one commodity in one step. This thesis work also considers an alternative disaggregate approach, where only one commodity is handled at a time. Finally, algorithms proposed for circuit switching network design problem are adapted to the packet switching case and an extensive comparative computational study is performed to point out the best method with respect to time and solution quality for a number of networks and cost structure. Computational results have shown that modification on Minoux greedy to eliminate cycles leads to considerable improvements and the disaggregate approach gives the best result in some networks and cost structure.

This master's thesis deals with optimization of traffic networks. There are discussed modeling methods for traffic and simplifications used in these models. Introductory part is focused on mathematical theory used to buid presented model. Main focus of this thesis was creation and optimization of the model which describes real world traffic network with traffic lights. Model was tested on both artificial and real data gathered in city Zlín. It was necessary to create generator which can produce suitable input data for model.

[pt] Esta dissertação apresenta novos algoritmos aproximados e uma abordagem exata para a resolução de um problema de correspondência inexata de grafos. O problema considerado é o de correspondência entre um grafo representando um modelo genérico e outro representando dados a serem reconhecidos. Assumi-se que o grafo dos dados possui mais vértices que o do modelo. A motivação para o estudo desse problema vem de problemas de reconhecimento de cenas, que consistem na caracterização dos objetos envolvidos em uma determinada cena, assim como das relações existentes entre eles. Uma aplicação para este problema na área de reconhecimento de imagens médicas é a de efetuar-se o reconhecimento de estruturas 3D do cérebro humano, a partir de imagens obtidas por ressonância magnética. Tais imagens são previamente processadas por algum método de segmentação automática e o processo de reconhecimento consiste na busca da correspondência estrutural entre a imagem e um modelo genérico, tipicamente definido como um atlas de imagens médicas. Foram propostos novos algoritmos aproximados, tais como um algoritmo construtivo guloso aleatorizado, um procedimento de reconexão de caminhos e um GRASP que combina estes com uma técnica de busca local. Além disso, foi proposta uma formulação original do problema como um problema de programação linear inteira, que permitiu a resolução de algumas instâncias de forma exata. / [en] This dissertation presents new approximation algorithms and an exact approach to the solution of an inexact graph matching problem. The problem consists in finding the best match between a generic model graph and a graph representing an image, the latter with more nodes than the former. The motivation for studying this problem comes from a scene recognition problem, which consists in characterizing objects involved in a given scene and the relationships between them. An application of this problem appears in the analysis of medical images and consists in recognizing 3-dimensional structures in the human brain using images obtained by magnetic resonance. Such images must be previously processed by an automatic segmentation method and the recognition process consists in the search of an structural matching between the image and a generic model, typically defined as an atlas of medical images. New heuristics are proposed, such as a greedy randomized construction algorithm, a path relinking procedure and a GRASP heuristic that combines them with a local search technique. Furthermore, an original integer formulation of the problem based on integer multicommodity flows is proposed, which makes possible the exact solution of medium- sized instances.

Η δρομολόγηση και ανάθεση μηκών κύματος (routing and wavelength assignment - RWA) αποτελεί ένα πολύ σημαντικό πρόβλημα, που απασχολεί τους σχεδιαστές WDM οπτικών δικτύων και είναι γνωστό, πως είναι NP-πλήρες. Στην εργασία αυτή σχεδιάζουμε και υλοποιούμε έναν αλγόριθμο για το στατικό RWA, που βασίζεται σε έναν προτεινόμενο σχηματισμό (μη ακέραιου) γραμμικού προγραμματισμού (linear programming - LP). Ισχυριζόμαστε, πως ο σχηματισμός αυτός είναι σε θέση να παρέχει ακέραιες βέλτιστες λύσεις (παρά την εν γένει μη ακέραια φύση του) για ένα μεγάλο ποσοστό στιγμιότυπων εισόδου, οδηγώντας έτσι σε αντίστοιχες ακριβείς λύσεις του RWA. Η πολυπλοκότητα του αλγόριθμου κυριαρχείται από το χρόνο εκτέλεσης του αλγόριθμου Simplex, ο οποίος θεωρείται αποδοτικός για μια μεγάλη πλειοψηφία στιγμιότυπων εισόδου. Στα διαφανή (πλήρως οπτικά) δίκτυα, η ποιότητα του σήματος υπόκειται σε μια ποικιλία από φυσικές εξασθενήσεις, όπως είναι η διασπορά λειτουργίας πόλωσης (polarization mode dispersion - PMD), ο θόρυβος αυθόρμητης εκπομπής ενισχυτή (amplified spontaneous emission - ASE - noise) και η χρωματική διασπορά (chromatic dispersion - CD). Αυτές οι εξασθενήσεις μοντελοποιούνται γραμμικά και μπορούν να αντιμετωπιστούν αποτελεσματικά από ένα σύνολο αναλυτικών τύπων ως επιπρόσθετοι περιορισμοί στο RWA. Εφαρμόζουμε τον αλγόριθμό μας και εκτελούμε RWA βασισμένο σε περιορισμούς εξασθένησης, με σκοπό να παρατηρήσουμε συγκριτικά αποτελέσματα στην απόδοση ενός τυπικού μητροπολιτικού δικτύου υπό διάφορες παραμέτρους του δικτύου και των εξασθενήσεων, όπως είναι ο ρυθμός bit, ο τύπος και το κέρδος των ενισχυτών, η χρησιμοποιούμενη διάταξη διαμόρφωσης, κλπ. / Routing and wavelength assignment (RWA) is a very important problem concerning WDM optical network designers and is known to be NP-complete. In this work, we design and implement an algorithm for the static RWA, that is based on a proposed (not integer) linear programming formulation. We claim, that this formulation is able to provide integer optimal solutions (despite its non integral nature) for a large fraction of input instances, yielding thus to corresponding exact RWA solutions. The algorithm's complexity is dominated by the execution time of Simplex LP-solver, that is considered efficient in the great majority of all possible input instances. In transparent (all-optical) networks, the signal quality is subject to a variety of physical impairments, such as polarization mode dispersion (PMD), amplified spontaneous emission (ASE) noise and chromatic dispersion (CD). Those impairments are linearly modeled and are handled effectively by a set of analytical formulae as additional constraints on RWA. We apply our algorithm to perform impairment-constraint based RWA, in order to obtain comparative results of a typical metropolitan network's performance under various network and impairment parameters, such as bit rate, amplifier gain and type, modulation format used, etc.