Дисертації з теми "Polyhedral approaches"

Cette thèse étudie les aspects polyhédraux de certains problèmes de conception de réseau, en se concentrant principalement sur les aspects liés à la connectivité dessous-structures nécessaires pour créer des applications réseau fiables. Le cœur de nombreuses applications différentes de conception de réseau réside dans le fait qu’il est nécessaire de fournir un sous-réseau connexe (pouvant être compris comme un ensemble de sommets ou d’arêtes induisant un sous-graphe connecté) pouvant présenter d’autres propriétés souhaitables, comme atteindre un certain niveau de capacité de survie ou de robustesse, des contraintes de capacité ou d’autres types de contraintes budgétaires, en fonction du contexte. La plupart des études menées et des algorithmes développés tentent de tirer parti de ces aspects particuliers qui différencient une application de l’autre, sans se préoccuper des aspects qui réunissent ces questions. Par conséquent, ce travail tente de développer une approche unifiée capable d’explorer les aspects les plus pertinents des problèmes de conception de réseau, en espérant que cela conduirait à une compréhension réfléchie de problèmes plus spécifiques, en apportant une contribution précieuse à la recherche
This theses study the polyhedral aspects of some network design problems, focusing most on the aspects related to connectivity of the substructures necessary to build reliable network applications. At theheart of many different network design applications lies the fact that one must provide a connected subnetwork (which can be viewed as a collection of vertices or edges inducing a connected subgraph) exhibiting other desirable properties, like achieving some level of survivability or robustness, capacity constraints,or other types of budgetary constraints, depending on the context.A majority of the studies conductedand of the algorithms developed tryto take advantage of those particular aspects that differentiate one application from another, and not much attention has been given to the aspectsthat bring together these questions. Most of the studies conducted and the algorithms developed try to take advantage of those particular aspects that differentiate one application from another, and not much attention has been given to the aspects that bring together these questions. Hence, this work tries to develop an unified approach capable of exploring the most pertinent aspects of network design problems hoping that this can lead to thoughtful insights to more specific problems, being a valuable contribution to the research community and it

NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches
Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

Polytopes $Q\sbsp{2E}{n}$ and $Q\sbsp{2N}{n}$, which are associated with the minimum cost 2-edge-connected subgraph problem and the minimum cost 2-node-connected subgraph problem, respectively, are studied in this thesis, and some new classes of facet-inducing inequalities are introduced for these polytopes. These classes of inequalities are related to the so-called clique tree inequalities for the travelling salesman polytope ($Q\sbsp{T}{n}$), and the relationships between $Q\sbsp{T}{n}$ and $Q\sbsp{2E}{n}, Q\sbsp{2N}{n}$ are exploited in obtaining these new classes of facets. Due to the use of problem specific facet-inducing inequalities instead of dominant cutting-planes, the linear programming cutting-plane method has proven to be quite successful for solving some NP-hard combinatorial optimization problems. We believe that our new classes of facet-inducing inequalities can be used to further improve the cutting-plane procedure for designing minimum cost survivable communication networks.

Applications that manipulate sparse data structures contain memory reference patterns that are unknown at compile time due to indirect accesses such as A[B[i]]. To exploit parallelism and improve locality in such applications, prior work has developed a number of Run-Time Reordering Transformations (RTRTs). This paper presents the Sparse Polyhedral Framework (SPF) for specifying RTRTs and compositions thereof and algorithms for automatically generating efficient inspector and executor code to implement such transformations. Experimental results indicate that the performance of automatically generated inspectors and executors competes with the performance of hand-written ones when further optimization is done.

In this thesis we explored the consequences of considering generalised non-quantum notions of entanglement in the classical simulation of noisy quantum computers where the available measurements are restricted. Such noise rates serve as upper bounds to fault tolerance thresholds. These measurement restrictions come about either through imperfection, and/or by design to some limited set. By considering sets of operators that return positive measurement outcome probabilities for the restricted measurements, one can construct new single particle state spaces containing quantum and non-quantum operators. These state spaces can then be used with a modified version of Harrow and Nielsen’s classical simulation algorithm to efficiently simulate noisy quantum computers that are incapable of generating generalised entanglement with respect to the new state spaces. Through this approach we developed alternative methods of classical simulation, strongly connected to the study of non-local correlations, in that we constructed noisy quantum computers capable of performing non-Clifford operations and could generate some forms of multiparty quantum entanglement, but were classical in that they could be efficiently classically simulated and could not generate non-local statistics. We focused on magic state quantum computers (that are limited to only Pauli measurements), with ideal local gates, but noisy control-Pauli Z gates, and calculated the noise needed to ensure the control-Z gates became incapable of generating generalised entanglment for a variety of noise models and state space choice, with the aim of finding an optimal single particle state space requiring the least noise to remove the generalised entanglement. The state spaces were required to always return valid measurement probabilities, this meant they also had had to have octahedral symmetry to ensure local gates did not take states outside the state space. While we able to find to the optimal choice for highly imperfect measurements, were we unable to find the optimal in all cases. Our best candidate state space required less joint depolarising noise at [approximately equal to] 56% in comparison to noise levels of [approximately equal to] 67% required if the algorithm used quantum notions of separability. This suggests that generalised entanglement may offer more insight than quantum entanglement when discussing the power of Clifford operation based quantum computers.

Dans cette thèse nous considérons deux problèmes d'optimisation combinatoire.Le premier s'appelle problème du double voyageur de commerce avec contraintes de piles. Dans ce problème, un véhicule doit ramasser un certain nombre d'objets dans une région pour les livrer à des clients situés dans une autre région. Lors du ramassage, les objets sont stockés dans les différentes piles du véhicule et la livraison des objets se fait selon une politique de type last-in-first-out. Le ramassage et la livraison consistent chacune en une tournée Hamiltonienne effectuée par le véhicule dans la région correspondante.Nous donnons une formulation linéaire en nombres entiers pour ce problème. Elle est basée sur des variables de précédence et sur des contraintes de chemins infaisables. Nous donnons par la suite des résultats polyédraux sur l'enveloppe convexe des solutions de notre formulation. En particulier, nous montrons des liens forts avec un polytope associé au problème du voyageur de commerce et des liens avec un polytope de type set covering. Cette étude polyédrale nous permet de renforcer la formulation initiale et de développer un algorithme de coupes et branchements efficace. Le deuxième problème que nous considérons consiste à trouver la description des polytopes lexicographiques. Ces derniers sont les enveloppes convexes des points entiers lexicographiquement compris entre deux points entiers fixés. Nous donnons une description complète de ces polytopes en termes d'inégalités linéaires. Nous démontrons que la famille des polytopes lexicographiques est fermée par intersection
In this thesis we consider two problems arising in combinatorial optimization.The first one is the double traveling salesman problem with multiple stacks. In this problem a vehicle picks up a given set of items in a region and subsequently delivers them to demanding customers in another region. When an item is picked up, it is put in a stack of the vehicle. The items are delivered observing a last-in-first-out policy. The pickup phase and the delivery phase consist in two Hamiltonian circuits, each performed by the vehicle in the corresponding region. We give a new integer linear programming formulation for this problem. Its main features are the presence of precedence variables and new infeasible path constraints. We provide polyhedral results on the convex hull of the solutions to our formulation. In particular, we show strong links with a specific TSPpolytope and a specific set covering polytope. We deduce strengthening inequalities for the initial formulation, subsequently embedded in an efficient branch-and-cut algorithm. The second problem we consider consists in finding the description of the lexicographical polytopes. These are convex hulls of the integer points lexicographically between two given integer points. We give a complete description of these polytopes by means of linear inequalities. We show that the lexicographical polytope family is closed under intersection

Malgré des décennies de recherche sur l’optimisation de boucle auxhaut niveau et leur intégration réussie dans les compilateurs C/C++et FORTRAN, la plupart des systèmes de transformation de bouclene traitent que partiellement les défis posé par la complexité croissanteet la diversité du matériel d’aujourd’hui. L’exploitation de laconnaissance dédiée a un domaine d’application pour obtenir le codeoptimal pour cibles complexes, tels que des accélérateurs ou des microprocessorsmulti-coeur, pose des problèmes pour les formalismeset outils d’optimisation de boucle existants. En conséquence, de nouveauxschémas d’optimisation qui exploitent la connaissance dédiéea un domaine sont développées indépendamment sans profiter dela technologie d’optimisation de boucle existante. Cela conduit à despossiblités d’optimisation raté et ainsi qu’à une faible portabilité deces schémas d’optimisation entre des compilateurs différents. Un domainepour lequel on voit la nécessité d’améliorer les optimisationsest le calcul de pochoir itératifs, un probléme de calcul important quiest réguliérement optimisé par les compilateurs dédiées, mais pourlequel générer code efficace est difficile.Dans ce travail, nous présentons des nouvelles stratégies pour l’optimisationdédiée qui permettent la génération de code GPU haute performancepour des calculs de pochoir. À la différence de la façon dontla plupart des compilateurs existants sont mis en oeuvre, nous découplonsla stratégie d’optimisation de haut niveau de l’optimisationde bas niveau et la spécialisation nécessaire pour obtenir la performanceoptimale. Comme schéma d’optimisation de haut niveau, nousprésentons une nouvelle formulation de “split tiling”, une techniquequi permet la réutilisation de données dans la dimension du tempsainsi que le parallélisme équilibré à gros grain sans la nécessité derecourir à des calculs redondants. Avec le “split tiling”, nous montronscomment intégrer une optimisation dédiée dans un traducteurgénérique source-à-source, C vers CUDA, une approche qui nouspermet de réutiliser des optimisations existants non-dédiées. Nousprésentons ensuite notre technique appelée “hybrid hexagonal / parallelogramtiling", un schéma qui nous permet de générer du codeque cible directement les préoccupations spécifiques aux GPUs. Pourconclure notre travail sur le "loop tiling", nous étudions la rapport entre“diamond tiling” et “hexagonal tiling”. À partir d’une analyse de“diamond tiling” détailée, qui comprend les exigences qu’elle posesur la taille de tuile et les coefficients de front d’onde, nous fournissonsune formulation unifiée de l’“hexagonal tiling” et du “diamondtiling” qui nous permet de réaliser un “hexagonal tiling” pourvdes problèmes avec deux dimensions (un temps, un espace) dans lecadre d’un usage dans un optimiseur générique, comme “Pluto”. Enfin,nous utilisons cette formulation pour évaluer l’“hexagonal tiling”et le “diamond tiling” en terme de rapport de calcul-à-communicationet calcul-à-synchronisation.Dans la deuxième partie de ce travail, nous discutons nos contributionsaux composants de l’infrastructure les plus important, nos“building blocks”, qui nous permettent de découpler notre optimisationde haut niveau tant des optimisations nécessaires dàns la générationde code que de l’infrastructure de compilation générique. Nouscommençons par présenter le nouveau “polyhedral extractor” (pet),qui obtient une représentation polyédrique d’un morceau de code C.pet utilise l’arithmétique de Presburger en sa généralité pour élargirle fragment de code C supporté et porter une attention particulièreà la modélisation de la sémantique des langages même en présencede dépassement de capacité des entiers
Despite decades of research on high-level loop optimizations and theirsuccessful integration in production C/C++/FORTRAN com- pilers, most compilerinternal loop transformation systems only partially address the challengesposed by the increased complexity and diversity of today’s hardware. Especiallywhen exploiting domain specific knowledge to obtain optimal code for complextargets such as accelerators or many-cores processors, many existing loopoptimization frameworks have difficulties exploiting this hardware. As aresult, new domain specific optimization schemes are developed independentlywithout taking advantage of existing loop optimization technology. This resultsboth in missed optimization opportunities as well as low portability of theseoptimization schemes to different compilers. One area where we see the need forbetter optimizations are iterative stencil computations, an importantcomputational problem that is regularly optimized by specialized, domainspecific compilers, but where generating efficient code is difficult.In this work we present new domain specific optimization strategies that enablethe generation of high-performance GPU code for stencil computations. Differentto how most existing domain specific compilers are implemented, we decouple thehigh-level optimization strategy from the low-level optimization andspecialization necessary to yield optimal performance. As high-leveloptimization scheme we present a new formulation of split tiling, a tilingtechnique that ensures reuse along the time dimension as well as balancedcoarse grained parallelism without the need for redundant computations. Usingsplit tiling we show how to integrate a domain specific optimization into ageneral purpose C-to-CUDA translator, an approach that allows us to reuseexisting non-domain specific optimizations. We then evolve split tiling into ahybrid hexagonal/parallelogram tiling scheme that allows us to generate codethat even better addresses GPU specific concerns. To conclude our work ontiling schemes we investigate the relation between diamond and hexagonaltiling. Starting with a detailed analysis of diamond tiling including therequirements it poses on tile sizes and wavefront coefficients, we provide aunified formulation of hexagonal and diamond tiling which enables us to performhexagonal tiling for two dimensional problems (one time, one space) in thecontext of a general purpose optimizer such as Pluto. Finally, we use thisformulation to evaluate hexagonal and diamond tiling in terms ofcompute-to-communication and compute-to-synchronization ratios.In the second part of this work, we discuss our contributions to importantinfrastructure components, our building blocks, that enviable us to decoupleour high-level optimizations from both the necessary code generationoptimizations as well as the compiler infrastructure we apply the optimizationto. We start with presenting a new polyhedral extractor that obtains apolyhedral representation from a piece of C code, widening the supported C codeto exploit the full generality of Presburger arithmetic and taking special careof modeling language semantics even in the presence of defined integerwrapping. As a next step, we present a new polyhedral AST generation approach,which extends AST generation beyond classical control flow generation byallowing the generation of user provided mappings. Providing a fine-grainedoption mechanism, we give the user fine grained control about AST generatordecisions and add extensive support for specialization e.g., with a newgeneralized form of polyhedral unrolling. To facilitate the implementation ofpolyhedral transformations, we present a new schedule representation, scheduletrees, which proposes to make the inherent tree structure of schedules explicitto simplify the work with complex polyhedral schedules.The last part of this work takes a look at our contributions to low-levelcompilers

Doctor en Sistemas de Ingeniería
En esta tesis se presentan los resultados del trabajo desarrollado por el autor durante el periodo en que fue estudiante de doctorado en el Departamento de Industrias de la Universidad de Chile. El trabajo se centra en la aplicación de técnicas de optimización entera-mixtas a problemas de planificación estratégica del sector eléctrico, donde el problema de corto plazo correspondiente al predespacho de unidades de generación en sistemas térmicos es el tema central en estudio. En lo relativo al modelamiento del problema de predespacho de unidades, se considera el análisis de las distintas formulaciones entera-mixtas disponibles en la literatura junto con una nueva basada en un formulaciones extendidas tipo red. Se investiga su desempeño sobre un conjunto de instancias reales desde el punto de vista de su eficiencia computacional al ser resueltas con softwares comerciales. Lo anterior incluye análisis de tiempos de solución, nodos utilizados e iteraciones de simplex realizadas para distintas tolerancias requeridas. Los experimentos muestran la calidad de la aproximación propuesta, siendo esta completamente competitiva respecto a las ya documentadas. Este resultado era esperable, dada la estructura totalmente unimodular de gran parte de la formulación propuesta, pero para nada justificable debido al tamaño de la misma. Lo anterior muestra que el efecto del preproceso de los softwares comerciales puede ser fundamental en algunas formulaciones. Por otro lado, respecto a la función objetivo del problema de predespacho de unidades, que por lo general se representa como una función cuadrática de la generación, se presenta una nueva manera de linealizar su comportamiento de modo que su inclusión en una formulación entera-mixta lineal tradicional sea eficiente. Esto último debe entenderse a partir de la necesidad que el tamaño de la aproximación no crezca de manera desmedida si el error requerido para la misma decrece. Si bien ya existía la posibilidad de hacer esto mediante la aplicación de la aproximación desarrollada por Ben-Tal y Nemirovsky para conos de segundo orden [2], acá se presenta un método alternativo, con mejores propiedades numéricas, un orden de magnitud mejor en calidad de aproximación, y cuya aplicación a problemas reales de predespacho de unidades genera mejores resultados respecto de las aproximaciones tradicionales. Por último, con el fin de mejorar el desempeño de la formulación entera-mixta presentada, se realiza el análisis poliedral de una de sus subestructuras esperando identificar desigualdades válidas que permitan mejorar su cota dual. Esta subestructura corresponde al knapsack semicontinuo con restricciones adicionales del tipo generalized upper bound. Se demuestra que bajo supuestos simples es posible identificar facetas tipo generalized flow cover en espacios restringidos de dimensión inferior. Luego se llevan estas desigualdades al espacio original utilizando procedimientos de lifting multidimensional independiente de la secuencia [38, 27, 16, 17] y se iii prueba que con supuestos adicionales también son facetas allí. Experimentos computacionales en instancias derivadas de problemas de UC muestran su eficiencia, donde más de un 50% del gap integral del nodo raíz se reduce aplicando en promedio solo tres de estos cortes. Además, en este contexto, también se ha implementado un solver ad-hoc para la solución eficiente de las relajaciones lineales de la formulación tipo red, con un speed-up del orden de 4x a 8x respecto a CPLEX barrier optimizer, pero que aún no está documentado.

In this thesis, we introduce the Partitioning-Hub Location-Routing problem (PHLRP), which can be classified as a variant of the hub location problem.

PHLRP consists of partitioning a network into sub-networks, locating at least one hub in each subnetwork and routing the traffic within the network such that all inter-subnetwork traffic is routed through the hubs and all intra-subnetwork traffic stays within the sub-networks all the way from the source to the destination. Obviously, besides the hub location component, PHLRP also involves a graph partitioning component and a routing component. PHLRP finds applications in the strategic planning or deployment of the Intermediate System-Intermediate System (ISIS) Internet Protocol networks and the Less-than-truck load freight distribution systems.

First, we introduce three IP formulations for solving PHLRP. The hub location component and the graph partitioning components of PHLRP are

modeled in the same way in all three formulations. More precisely, the hub location component is represented by the p-median variables and constraints; and the graph partitioning component is represented by the size-constrained graph partitioning variables and constraints. The formulations differ from each other in the way the peculiar routing requirements of PHLRP are modeled.

We then carry out analytical and empirical comparisons of the three IP

formulations. Our thorough analysis reveals that one of the formulations is

provably the tightest of the three formulations. We also show analytically that the LP relaxations of the other two formulations do not dominate each other. On the other hand, our empirical comparison in a standard branch-and-cut framework that is provided by CPLEX shows that not the tightest but the most compact of the three formulations yield the best performance in terms of solution time.

From this point on, based on the insight gained from detailed analysis of the formulations, we focus our attention on a common sub-problem of the three formulations: the so-called size-constrained graph partitioning problem. We carry out a detailed polyhedral analysis of this problem. The main benefit from this polyhedral analysis is that the facets we identify for the size-constrained graph partitioning problem constitute strong valid inequalities for PHLRP.

And finally, we wrap up our efforts for solving PHLRP. Namely, we present

the results of our computational experiments, in which we employ some facets

of the size-constrained graph partitioning polytope in a branch-and-cut algorithm for solving PHLRP. Our experiments show that our approach brings

significant improvements to the solution time of PHLRP when compared with

the default branch-and-cut solver of XPress.

/

Dans cette thèse, nous introduisons le problème Partitionnement-Location des Hubs et Acheminement (PLHA), une variante du problème de location de hubs. Le problème PLHA partitionne un réseau afin d'obtenir des sous-réseaux, localise au moins un hub dans chaque sous-réseau et achemine le traffic dans le réseau de la maniére suivante :le traffic entre deux

sous-réseaux distincts doit être éxpedié au travers des hubs tandis que le traffic entre deux noeuds d'un même sous-réseau ne doit pas sortir de celui-ci. PLHA possède des applications dans le planning stratégique, ou déploiement, d'un certain protocole de communication utilisé

dans l'Internet, Intermediate System - Intermediate System, ainsi que dans la distribution des frets.

Premièrement, nous préesentons trois formulations linéaires en variables entières pour résoudre PLHA. Le partitionnement du graphe et la localisation des hubs sont modélisées de la même maniére dans les trois formulations. Ces formulations diffèrent les unes des autres dans la maniére dont l'acheminement du traffic est traité.

Deuxièmement, nous présentons des comparaisons analytiques et empiriques des trois formulations. Notre comparaison analytique démontre que l'une des formulations est plus forte que les autres. Néanmoins, la comparaison empirique des formulations, via le solveur CPLEX, montre que la formulation la plus compacte (mais pas la plus forte) obtient les meilleures performances en termes de temps de résolution du problème.

Ensuite, nous nous concentrons sur un sous-problème, à savoir, le partitionnement des graphes sous contrainte de taille. Nous étudions le polytope des solutions réalisables de ce sous-problème. Les facettes de ce polytope constituent des inégalités valides fortes pour

PLHA et peuvent être utilisées dans un algorithme de branch-and-cut pour résoudre PLHA.

Finalement, nous présentons les résultats d'un algorithme de branch-and-cut que nous avons développé pour résoudre PLHA. Les résultats démontrent que la performance de notre méthode est meilleure que celle de l'algorithme branch-and-cut d'Xpress.

Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Considérons un graphe G = (V,E,w) non orienté dont les sommets sont pondérés et un entier k. Le problème à étudier consiste à la construction des algorithmes afin de déterminer le nombre minimum de nœuds qu’il faut enlever au graphe G pour que toutes les composantes connexes restantes contiennent chacune au plus k-sommets. Ce problème nous l’appelons problème de k-Séparateur et on désigne par k-séparateur le sous-ensemble recherché. Il est une généralisation du Vertex Cover qui correspond au cas k = 1 (nombre minimum de sommets intersectant toutes les arêtes du graphe)
Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problem

Dans cette thèse, nous proposons une nouvelle approche de gestion de risques pour les réseaux de télécommunications. Celle-ci est basée sur le concept de graphes d’analyse de risques appelés Risk Assessment Graphs (RAGs). Ces graphes contiennent deux types de noeuds : des points d’accés qui sont des points de départ pour les attaquants, et des noeuds appelés bien-vulnérabilité. Ces derniers doivent être sécurisés. La propagation potentielle d’un attaquant entre deux noeuds est représentée par un arc dans le RAG. Un poids positif représentant la difficulté de propagation d’un attaquant est associé à chaque arc. D’abord, nous proposons une approche quantitative d’évaluation de risques basée sur le calcul des plus courts chemins entre les points d’accés et les noeuds bien-vulnérabilité. Nous considérons ensuite un problème de traitement de risque appelé Proactive Countermeasure Selection Problem (PCSP). Etant donnés un seuil de difficulté de propagation pour chaque paire de point d’accés et noeud bien-vuln ́erabilité, et un ensemble de contremesures pouvant être placées sur les noeuds bien-vulnérabilité, le problème PCSP consiste à déterminer le sous ensemble de contremesures de coût minimal, de manière à ce que la longueur de chaque plus court chemin d’un point d’accés à un noeud bien-vulnérabilité soit supérieure ou égale au seuil de difficulté de propagation. Nous montrons que le PCSP est NP-complet même quand le graphe est réduit à un arc. Nous donnons aussi une formulation du problème comme un modèle de programmation bi-niveau pour lequel nous proposons deux reformulations en un seul niveau: une formulation compacte basée sur la dualité en programmation linéaire, et une formulation chemins avec un nombre exponentiel de contraintes, obtenue par projection. Nous étudions cette deuxième formulation d’un point de vue polyhèdral. Nous décrivons différentes classes d’inégalités valides. Nous discutons l’aspect facial des inégalités de base et des inégalités valides. Nous concevons aussi des méthodes de séparation pour ces inégalités. En utilisant ces résultats, nous développons un algorithme de coupes et branchements pour le problème. Nous discutons enfin d’une étude numérique approfondie montrant l'éfficacité des résultats polyhèdraux d’un point de vue algorithmique. Notre approche s’applique à une large gamme de cas réels dans le domaine de télécommunications. Nous l’illustrons à travers plusieurs cas d’utilisation couvrant l’internet des objets (IoT), les réseaux orient ́es logiciel (SDN) et les réseaux locaux (LANs). Aussi, nous montrons l’intégration de notre approche dans une application web
In this thesis, we propose a new risk management framework for telecommunication networks. This is based on theconcept of Risk Assessment Graphs (RAGs). These graphs contain two types of nodes: access point nodes, or startingpoints for attackers, and asset-vulnerability nodes. The latter have to be secured. An arc in the RAG represents apotential propagation of an attacker from a node to another. A positive weight, representing the propagation difficulty ofan attacker, is associated to each arc. First, we propose a quantitative risk evaluation approach based on the shortestpaths between the access points and the asset-vulnerability nodes. Then, we consider a risk treatment problem, calledProactive Countermeasure Selection Problem (PCSP). Given a propagation difficulty threshold for each pair of accesspoint and asset-vulnerability node, and a set of countermeasures that can be placed on the asset vulnerability nodes, thePCSP consists in selecting the minimum cost subset of countermeasures so that the length of each shortest path froman access point to an asset vulnerability node is greater than or equal to the propagation difficulty threshold.We show that the PCSP is NP-Complete even when the graph is reduced to an arc. Then, we give a formulation of theproblem as a bilevel programming model for which we propose two single-level reformulations: a compact formulationbased on LP-duality, and a path formulation with an exponential number of constraints, obtained by projection. Moreover,we study the path formulation from a polyhedral point of view. We introduce several classes of valid inequalities. Wediscuss when the basic and valid inequalities define facets. We also devise separation routines for these inequalities.Using this, we develop a Branch-and-Cut algorithm for the PCSP along with an extensive computational study. Thenumerical tests show the efficiency of the polyhedral results from an algorithmic point of view.Our framework applies to a wide set of real cases in the telecommunication industry. We illustrate this in several practicaluse cases including Internet of Things (IoT), Software Defined Network (SDN) and Local Area Networks (LANs). We alsoshow the integration of our approach in a web application

NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches.
Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze.

This study was carried out to investigate the effect of using animated computer 3-D figures illustration (ACTDFI) in the learning of polyhedron in geometry. By random sampling, intact group of four grade 9 classes in four different schools from a cluster of four educational district schools of Limpopo province in South Africa were selected. The study involved quasi-experimental and inquiry research approaches, the quasi-experimental approach involved pre and posttest design while the inquiry research approach involve classroom observation. There were three experimental groups and a control group with a total of 174 study participants. ACTDFI was used as an intervention for two weeks in the three experimental groups while in the control group, chalk-talk traditional teaching approach was used. Pre-test and post-test was used to collect quantitative data while classroom observation was used to collect qualitative data. This study was carried out to investigate the effect of using animated computer 3-D figures illustration (ACTDFI) in the learning of polyhedron in geometry. By random sampling, intact group of four grade 9 classes in four different schools from a cluster of four educational district schools of Limpopo province in South Africa were selected. The study involved quasi-experimental and inquiry research approaches, the quasi-experimental approach involved pre and posttest design while the inquiry research approach involve classroom observation. There were three experimental groups and a control group with a total of 174 study participants. ACTDFI was used as an intervention for two weeks in the three experimental groups while in the control group, chalk-talk traditional teaching approach was used. Pre-test and post-test was used to collect quantitative data while classroom observation was used to collect qualitative data. The findings from the quantitative Classroom observations were carried out to collect relevant data on how the study participants were taught stationary points in differential calculus, especially with the use of the constructivist pedagogical approach. A suitable observation checklist was developed for this purpose (Appendix 6 refers). Classroom observation checklist is a list of factors to be considered while observing a class. It gives a structure and framework for the observation. suggested that the use of ACTDFI might have improved academic achievement in learning of polyhedron during the intervention, while the qualitative data analysis indicated that the use of ACTDFI in the experimental groups might have facilitated the learning of the concepts of polyhedron. It is therefore recommended that further research is necessary on the application of ACTDFI in the teaching of 3-dimensional shapes at the primary schools
Mathematics Education
M. Sc. (Mathematics Education)

Geometric Multigrid (GMG) methods are widely used in numerical analysis to accelerate the convergence of partial differential equations solvers using a hierarchy of grid discretizations. These solvers find plenty of applications in various fields in engineering and scientific domains, where solving PDEs is of fundamental importance. Using multigrid methods, the pace at which the solvers arrive at the solution can be improved at an algorithmic level. With the advance in modern computer architecture, solving problems with higher complexity and sizes is feasible - this is also the case with multigrid methods. However, since hardware support alone cannot achieve high performance in execution time, there is a need for good software that help programmers in doing so. Multiple grid sizes and recursive expression of multigrid cycles make the task of manual program optimization tedious and error-prone. A high-level language that aids domain experts to quickly express complex algorithms in a compact way using dedicated constructs for multigrid methods and with good optimization support is thus valuable. Typical computation patterns in a GMG algorithm includes stencils, point-wise accesses, restriction and interpolation of a grid. These computations can be optimized for performance on modern architectures using standard parallelization and locality enhancement techniques. Several past works have addressed the problem of automatic optimizations of computations in various scientific domains using a domain-specific language (DSL) approach. A DSL is a language with features to express domain-specific computations and compiler support to enable optimizations specific to these computations. Halide and PolyMage are two of the recent works in this direction, that aim to optimize image processing pipelines. Many computations like upsampling and downsampling an image are similar to interpolation and restriction in geometric multigrid methods. In this thesis, we demonstrate how high performance can be achieved on GMG algorithms written in the PolyMage domain-specific language with new optimizations we added to the compiler. We also discuss the implementation of non-trivial optimizations, on PolyMage compiler, necessary to achieve high parallel performance for multigrid methods on modern architectures. We realize these goals by: • introducing multigrid domain-specific constructs to minimize the verbosity of the algorithm specification; • storage remapping to reduce the memory footprint of the program and improve cache locality exploitation; • mitigating execution time spent in data handling operations like memory allocation and freeing, using a pool of memory, across multiple multigrid cycles; and • incorporating other well-known techniques to leverage performance, like exploiting multi-dimensional parallelism and minimizing the lifetime of storage buffers. We evaluate our optimizations on a modern multicore system using five different benchmarks varying in multigrid cycle structure, complexity and size, for two-and three-dimensional data grids. Experimental results show that our optimizations: • improve performance of existing PolyMage optimizer by 1.31x; • are better than straight-forward parallel and vector implementations by 3.2x; • are better than hand-optimized versions in conjunction with optimizations by Pluto, a state-of-the-art polyhedral source-to-source optimizer, by 1.23x; and • achieve up to 1.5$\times$ speedup over NAS MG benchmark from the NAS Parallel Benchmarks. (The speedup numbers are Geometric means over all benchmarks)