Robust Design of Single-Commodity Networks

The results in the present work were obtained in a collaboration with Eduardo Álvarez- Miranda, Valentina Cacchiani, Tim Dorneth, Michael Jünger, Frauke Liers, Andrea Lodi and Tiziano Parriani. The subject of this thesis is a robust network design problem, i.e., a problem of the type “dimension a network such that it has sufficient capacity in all likely scenarios.” In our case, we model the network with an undirected graph in which each scenario defines a supply or demand for each node. We say that a flow in the network is feasible for a scenario if it can balance out its supplies and demands. A scenario polytope B defines which scenarios are relevant. The task is now to find integer capacities that minimize the total installation costs while allowing for a feasible flow in each scenario. This problem is called Single-Commodity Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and Sanità (INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph and a terminal set, find a minimum cost subtree that connects all terminals) and therefore is N P-hard. The problem is also a natural extension of minimum cost flows. The network design literature treats the case that the scenario polytope B is given as the finite set of its extreme points (finite case) and that it is given as the feasible region of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent, however, an efficient transformation is not possible in general. Buchheim, Liers and Sanità (INOC 2011) propose a Branch-and-Cut algorithm for the finite case. In this case, there exists a canonical problem formulation as a mixed integer linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers and Sanità enhance the formulation with general cutting planes that are called target cuts. The first part of the dissertation considers the problem variant where every scenario has exactly two terminal nodes. If the underlying network is a complete, unweighted graph, then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl. Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special case. However, these algorithms are based on the fact that complete graphs are Hamiltonian. The result of this part is a similar algorithm for hypercube graphs that assumes a special distribution of the supplies and demands. These graphs are also Hamiltonian. The second part of the thesis discusses the structure of the polyhedron of feasible sRND solutions. Here, the first result is a new MIP-based capacity formulation for the sRND problem. The size of this formulation is independent of the number of extreme points of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called cut-set inequalities that are known in similar form from other network design problems. By adapting a proof by Mattia (Computational Optimization and Applications 2013), we show that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear programming relaxation of the capacity formulation, we interpret certain general mixed integer cuts as 3-partition inequalities and show that these inequalities induce facets as well. The capacity formulation has exponential size and we therefore need a separation algorithm for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to a minimum cut problem that can be solved in polynomial time. In the polyhedral case, however, the separation problem is N P-hard, even if we assume that the scenario polytope is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can solve the separation problem in practice: We show a MIP based separation procedure for the Hose scenario polytope. Additionally, the thesis presents two separation methods for 3-partition inequalities. These methods are independent of the encoding of the scenario polytope. Additionally, we present several rounding heuristics. The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the algorithm in the last part of the thesis. There, we show experimentally that the algorithm works in practice, both in the finite and in the polyhedral case. As a reference point, we use a CPLEX implementation of the flow based formulation and the computational results by Buchheim, Liers and Sanità. Our experiments show that the new Branch-and-Cut algorithm is an improvement over the existing approach. Here, the algorithm excels on problem instances with many scenarios. In particular, we can show that the MIP separation of the cut-set inequalities is practical

Study of manifold geometry using non-negative kernel graphs

Amb l'augment de la mida de les dades, els sistemes efectius de reducció de la dimensionalitat s'han tornat necessaris per una gran varietat de tasques. Un conjunt de dades es pot caracteritzar per les seves propietats geomètriques, entre les quals es troben la densitat dels punts que hi té, la seva curvatura, i la dimensionalitat. En aquest context, la dimensió intrínseca (ID) fa referència al nombre mínim de paràmetres necessaris per caracteritzar un conjunt de dades. S'han proposat moltes eines per a l'estimació de DI, i les que aconsegueixen els millors resultats estan molt enfocades a resoldre aquest objectiu. Aquests estimadors altament especialitzats no permeten la interpretació de la geometria local de les dades en altres aspectes a part de la ID. A més, els mètodes que si ho permeten no són capaços d'estimar la ID de manera fiable. Proposem l'ús de grafs de kernel no negatiu (NNK), una aproximació a la construcció de grafs que caracteritza la geometria local de les dades, per estudiar la dimensió i la forma de les superfícies mutlidimensionals de dades a múltiples escales. Proposem l'ús d'una sèrie de propietats relacionades amb els grafs NNK per obtenir informació sobre diversos conjunts de dades. En particular, observem el nombre de veïns en un graf NNK, la dimensió de les aproximacions per anàlisi de components principals tant per als grafs K-nearest neighbor (KNN) com NNK, el diàmetre dels polítops definits pels grafs NNK i els angles principals entre les aproximacions per anàlisi de components principals dels grafs NNK. A més, estudiem aquestes propietats a múltiples escales utilitzant un algorisme que fa que les dades siguin més disperses fusionant punts en funció d'una tria de similitud. Utilitzant una similitud basada en els conjunts de veïns NNK, podem submostrejar conjunts de dades preservant les propietats geomètriques del conjunt de dades inicial.Given the increasing amounts of data being measured and recorded, effective dimensionality reduction systems have become necessary for a wide variety of tasks. A dataset can be characterized by its geometrical properties, including its point density, curvature, and dimensionality. In this context, the intrinsic dimension (ID) refers to the minimum number of parameters required to characterize a dataset. Many tools have been proposed for the estimation of ID, and the ones that achieve the best results are narrowly focused on solving this goal. These highly specialized estimators don't allow for the interpretation of the local geometry of the data in other aspects besides ID. Moreover, methods that do make this possible are not able to estimate ID reliably. We propose the use of non-negative kernel (NNK) graphs, an approach to graph construction that characterizes the local geometry of the data, to study the dimension and shape of data manifolds at multiple scales. We propose the use of a series of properties related to NNK graphs to gain insight into manifold datasets. In particular, we look at the number of neighbors in an NNK graph, the dimension of the low-rank approximations for both K-nearest neighbor (KNN) and NNK graphs, the diameter of the polytopes defined by NNK graphs, and the principal angles between the low-rank approximations of NNK graphs. Moreover, we study these properties at multiple scales using an algorithm that makes data sparse by merging points based on a choice of similarity. By using similarity based on local NNK neighborhoods we can subsample datasets preserving the geometrical properties of the initial dataset

Depth Lower Bounds in Stabbing Planes for Combinatorial Principles

We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in the case of the Pigeonhole principle are tight. The technique known so far to prove depth lower bounds for Stabbing Planes is a generalization of that used for the Cutting Planes proof system. In this work we introduce two new approaches to prove length/depth lower bounds in Stabbing Planes: one relying on Sperner's Theorem which works for the Pigeonhole principle and Tseitin contradictions over the complete graph; a second proving the lower bound for Tseitin contradictions over a grid graph, which uses a result on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz

Additive structures and randomness in combinatorics

Arithmetic Combinatorics, Combinatorial Number Theory, Structural Additive Theory and Additive Number Theory are just some of the terms used to describe the vast field that sits at the intersection of Number Theory and Combinatorics and which will be the focus of this thesis. Its contents are divided into two main parts, each containing several thematically related results. The first part deals with the question under what circumstances solutions to arbitrary linear systems of equations usually occur in combinatorial structures..The properties we will be interested in studying in this part relate to the solutions to linear systems of equations. A first question one might ask concerns the point at which sets of a given size will typically contain a solution. We will establish a threshold and also study the distribution of the number of solutions at that threshold, showing that it converges to a Poisson distribution in certain cases. Next, Van der Waerden’s Theorem, stating that every finite coloring of the integers contains monochromatic arithmetic progression of arbitrary length, is by some considered to be the first result in Ramsey Theory. Rado generalized van der Waerden’s result by characterizing those linear systems whose solutions satisfy a similar property and Szemerédi strengthened it to a statement concerning density rather than colorings. We will turn our attention towards versions of Rado’s and Szemerédi’s Theorem in random sets, extending previous work of Friedgut, Rödl, Rucin´ski and Schacht in the case of the former and of Conlon, Gowers and Schacht for the latter to include a larger variety of systems and solutions. Lastly, Chvátal and Erdo¿s suggested studying Maker-Breaker games. These games have deep connections to the theory of random structures and we will build on work of Bednarska and Luczak to establish the threshold for how much a large variety of games need to be biased in favor of the second player. These include games in which the first player wants to occupy a solution to some given linear system, generalizing the van der Waerden games introduced by Beck. The second part deals with the extremal behavior of sets with interesting additive properties. In particular, we will be interested in bounds or structural descriptions for sets exhibiting some restrictions with regards to either their representation function or their sumset. First, we will consider Sidon sets, that is sets of integers with pairwise unique differences. We will study a generalization of Sidon sets proposed very recently by Kohayakawa, Lee, Moreira and Rödl, where the pairwise differences are not just distinct, but in fact far apart by a certain measure. We will obtain strong lower bounds for such infinite sets using an approach of Cilleruelo. As a consequence of these bounds, we will also obtain the best current lower bound for Sidon sets in randomly generated infinite sets of integers of high density. Next, one of the central results at the intersection of Combinatorics and Number Theory is the Freiman–Ruzsa Theorem stating that any finite set of integers of given doubling can be efficiently covered by a generalized arithmetic progression. In the case of particularly small doubling, more precise structural descriptions exist. We will first study results going beyond Freiman’s well-known 3k–4 Theorem in the integers. We will then see an application of these results to sets of small doubling in finite cyclic groups. Lastly, we will turn our attention towards sets with near-constant representation functions. Erdo¿s and Fuchs established that representation functions of arbitrary sets of integers cannot be too close to being constant. We will first extend the result of Erdo¿s and Fuchs to ordered representation functions. We will then address a related question of Sárközy and Sós regarding weighted representation function.La combinatòria aritmètica, la teoria combinatòria dels nombres, la teoria additiva estructural i la teoria additiva de nombres són alguns dels termes que es fan servir per descriure una branca extensa i activa que es troba en la intersecció de la teoria de nombres i de la combinatòria, i que serà el motiu d'aquesta tesi doctoral. La primera part tracta la qüestió de sota quines circumstàncies es solen produir solucions a sistemes lineals d’equacions arbitràries en estructures additives. Una primera pregunta que s'estudia es refereix al punt en que conjunts d’una mida determinada contindran normalment una solució. Establirem un llindar i estudiarem també la distribució del nombre de solucions en aquest llindar, tot demostrant que en certs casos aquesta distribució convergeix a una distribució de Poisson. El següent tema de la tesis es relaciona amb el teorema de Van der Waerden, que afirma que cada coloració finita dels nombres enters conté una progressió aritmètica monocromàtica de longitud arbitrària. Aquest es considera el primer resultat en la teoria de Ramsey. Rado va generalitzar el resultat de van der Waerden tot caracteritzant en aquells sistemes lineals les solucions de les quals satisfan una propietat similar i Szemerédi la va reforçar amb una versió de densitat del resultat. Centrarem la nostra atenció cap a versions del teorema de Rado i Szemerédi en conjunts aleatoris, ampliant els treballs anteriors de Friedgut, Rödl, Rucinski i Schacht i de Conlon, Gowers i Schacht. Per últim, Chvátal i Erdos van suggerir estudiar estudiar jocs posicionals del tipus Maker-Breaker. Aquests jocs tenen una connexió profunda amb la teoria de les estructures aleatòries i ens basarem en el treball de Bednarska i Luczak per establir el llindar de la quantitat que necessitem per analitzar una gran varietat de jocs en favor del segon jugador. S'inclouen jocs en què el primer jugador vol ocupar una solució d'un sistema lineal d'equacions donat, generalitzant els jocs de van der Waerden introduïts per Beck. La segona part de la tesis tracta sobre el comportament extrem dels conjunts amb propietats additives interessants. Primer, considerarem els conjunts de Sidon, és a dir, conjunts d’enters amb diferències úniques quan es consideren parelles d'elements. Estudiarem una generalització dels conjunts de Sidons proposats recentment per Kohayakawa, Lee, Moreira i Rödl, en que les diferències entre parelles no són només diferents, sinó que, en realitat, estan allunyades una certa proporció en relació a l'element més gran. Obtindrem límits més baixos per a conjunts infinits que els obtinguts pels anteriors autors tot usant una construcció de conjunts de Sidon infinits deguda a Cilleruelo. Com a conseqüència d'aquests límits, obtindrem també el millor límit inferior actual per als conjunts de Sidon en conjunts infinits generats aleatòriament de nombres enters d'alta densitat. A continuació, un dels resultats centrals a la intersecció de la combinatòria i la teoria dels nombres és el teorema de Freiman-Ruzsa, que afirma que el conjunt suma d'un conjunt finit d’enters donats pot ser cobert de manera eficient per una progressió aritmètica generalitzada. En el cas de que el conjunt suma sigui de mida petita, existeixen descripcions estructurals més precises. Primer estudiarem els resultats que van més enllà del conegut teorema de Freiman 3k-4 en els enters. Llavors veurem una aplicació d’aquests resultats a conjunts de dobles petits en grups cíclics finits. Finalment, dirigirem l’atenció cap a conjunts amb funcions de representació gairebé constants. Erdos i Fuchs van establir que les funcions de representació de conjunts arbitraris d’enters no poden estar massa a prop de ser constants. Primer estendrem el resultat d’Erdos i Fuchs a funcions de representació ordenades. A continuació, abordarem una pregunta relacionada de Sárközy i Sós sobre funció de representació ponderada

Introduction to Mathematical Programming-Based Error-Correction Decoding

Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman et al. At first celebrated mainly for its analytical powers, real-world applications of LP decoding are now within reach thanks to most recent research. This document gives an elaborate introduction into both mathematical optimization and coding theory as well as a review of the contributions by which these two areas have found common ground.Comment: LaTeX sources maintained here: https://github.com/supermihi/lpdintr

On some problems related to 2-level polytopes

In this thesis we investigate a number of problems related to 2-level polytopes, in particular from the point of view of the combinatorial structure and the extension complexity. 2-level polytopes were introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems ranging from information theory to semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter, we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the second chapter, we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain non-trivial lower and upper bounds. In the third chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fourth chapter we address the problem of explicitly obtaining small size extended formulations whose existence is guaranteed by communication protocols. In particular we give an algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols

Zuordnungsproblem auf Hypergraphen

Diese Arbeit beschäftigt sich mit dem Hypergraph Assignment Problem (Abkürzung "HAP", dt.: Zuordnungsproblem auf Hypergraphen), einem Mengenzerlegungsproblem auf einem speziellen Typ von Hypergraphen. Das HAP verallgemeinert das Zuordnungsproblem von bipartiten Graphen auf eine Struktur, die wir bipartite Hypergraphen nennen, und ist durch eine Anwendung in der Umlaufplanung im Schienenverkehr motiviert. Die Hauptresultate betreffen die Komplexität, polyedrische Ergebnisse, die Analyse von Zufallsinstanzen sowie primale Methoden für das HAP. Wir beweisen, dass das HAP NP-schwer und APX-schwer ist, sogar wenn wir uns auf kleine Hyperkantengrößen und Hypergraphen mit einer speziellen, partitionierten Struktur beschränken. Darüber hinaus untersuchen wir die Komplexität der Mengenpackungs- sowie Mengenüberdeckungsrelaxierung und geben für bestimmte Fälle Approximations- und exakte Algorithmen mit einer polynomiellen Laufzeit an. Für das Polytop des Zuordnungsproblems ist eine vollständige lineare Beschreibung bekannt. Wir untersuchen daher auch das HAP-Polytop. Dafür ist die Anzahl der Facettenungleichungen schon für sehr kleine Problemgrößen sehr groß. Wir beschreiben eine Methode zur Aufteilung der Ungleichungen in Äquivalenzklassen, die ohne die Verwendung von Normalformen auskommt. Die Facetten in jeder Klasse können durch Symmetrien ineinander überführt werden. Es genügt, einen Repräsentanten aus jeder Klasse anzugeben, um ein vollständiges Bild der Polytopstruktur zu erhalten. Wir beschreiben den Algorithmus "HUHFA", der diese Klassifikation nicht nur für das HAP, sondern für beliebige kombinatorische Optimierungsprobleme, die Symmetrien enthalten, durchführt. Die größtmögliche HAP-Instanz, für die wir die vollständige lineare Beschreibung berechnen konnten, hat 14049 Facetten, die in 30 Symmetrieklassen aufgeteilt werden können. Wir können 16 dieser Klassen kombinatorisch interpretieren. Dafür verallgemeinern wir Odd-Set-Ungleichungen für das Matchingproblem unter Verwendung von Cliquen. Die Ungleichungen, die wir erhalten, sind gültig für Mengenpackungsprobleme in beliebigen Hypergraphen und haben eine klare kombinatorische Bedeutung. Die Analyse von Zufallsinstanzen erlaubt einen besseren Einblick in die Struktur von Hyperzuordnungen. Eine solche ausführliche Analyse wurde in der Literatur theoretisch und praktisch bereits für das Zuordnungsproblem durchgeführt. Als eine Verallgemeinerung dieser Ergebnisse für das HAP beweisen wir Schranken für den Erwartungswert einer Hyperzuordnung mit minimalen Kosten, die genau die Hälfte der maximal möglichen Anzahl an Hyperkanten, die keine Kanten sind, benutzt. In einem sog. vollständigen partitionierten Hypergraphen G2,2n mit Hyperkantenkosten, die durch unabhängig identisch exponentiell verteilte Zufallsvariablen mit Erwartungswert 1 bestimmt sind, liegt dieser Wert zwischen 0.3718 und 1.8310, wenn die Knotenanzahl gegen unendlich strebt. Schließlich entwickeln wir eine exakte kombinatorische Lösungsmethode für das HAP, die drei Ansätze kombiniert: Eine Nachbarschaftssuche mit Nachbarschaften exponentieller Größe, die Composite-Columns-Methode für das Mengenzerlegungsproblem sowie den Netzwerksimplexalgorithmus.This thesis deals with the hypergraph assignment problem (HAP), a set partitioning problem in a special type of hypergraph. The HAP generalizes the assignment problem from bipartite graphs to what we call bipartite hypergraphs, and is motivated by applications in railway vehicle rotation planning. The main contributions of this thesis concern complexity, polyhedral results, analyses of random instances, and primal methods for the HAP. We prove that the HAP is NP-hard and APX-hard even for small hyperedge sizes and hypergraphs with a special partitioned structure. We also study the complexity of the set packing and covering relaxations of the HAP, and present for certain cases polynomial exact or approximation algorithms. A complete linear description is known for the assignment problem. We therefore also study the HAP polytope. There, we have a huge number of facet-defining inequalities already for a very small problem size. We describe a method for dividing the inequalities into equivalence classes without resorting to a normal form. Within each class, facets are related by certain symmetries and it is sufficient to list one representative of each class to give a complete picture of the structural properties of the polytope. We propose the algorithm "HUHFA" for the classification that is applicable not only to the HAP but combinatorial optimization problems involving symmetries in general. In the largest possible HAP instance for which we could calculate the complete linear description, we have 14049 facets, which can be divided into 30 symmetry classes. We can combinatorially interpret 16 of these classes. This is possible by employing cliques to generalize the odd set inequalities for the matching problem. The resulting inequalities are valid for the polytope associated with the set packing problem in arbitrary hypergraphs and have a clear combinatorial meaning. An analysis of random instances provides a better insight into the structure of hyperassignments. Previous work has extensively analyzed random instances for the assignment problem theoretically and practically. As a generalization of these results for the HAP, we prove bounds on the expected value of a minimum cost hyperassignment that uses half of the maximum possible number of hyperedges that are not edges. In a certain complete partitioned hypergraph G2,2n with i. i. d. exponential random variables with mean 1 as hyperedge costs it lies between 0.3718 and 1.8310 if the vertex number tends to infinity. Finally, we develop an exact combinatorial solution algorithm for the HAP that combines three methods: A very large-scale neighborhood search, the composite columns method for the set partitioning problem, and the network simplex algorithm

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem

Complexity Theory, Game Theory, and Economics: The Barbados Lectures

This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some recent citations to v1 Revised v3 corrects a few typos in v