Algebraic theory of affine monoids

This thesis treats several aspects of affine monoids. First, we consider the structure of the set of holes of an affine monoid Q. This set is the difference between Q and its normalization. We find connections to algebraic properties of the monoi algebra K[Q] and in particular to its local cohomology. In special cases, this allows to compute the depth of K[Q]. We then specialize to simplicial and seminormal affine monoids and reprove or extend several known results using our theory. Moreover, we consider the dependency of algebraic properties of K[Q] upon the field K. We can show that Serre's (S3) does not depend on K. Further, we prove a special case of a conjecture by Eisenbud and Goto. In the next chapter, we construct a fammily of non-normal lattice simplices, such that the holes have an arbitrary lattice distance from the facets. In the next chapter, we consider toric edge rings and give a criterion for the toric egde ring to satisfy Serre's (R1) and for being seminormal. In the last chapter, we consider the affine monoid of the linear ordering polytope. Using graph theory, we give a combinatorial description of the degree 2 generators of its toric ideal

Modelos para sequenciação de padrões em problemas de corte de stock

Tese de doutoramento em Engenharia Industrial e de SistemasIn this thesis, we address an optimization problem that appears in cutting stock operations research called the minimization of the maximum number of open stacks (MOSP) and we put forward a new integer programming formulation for the MOSP. By associating the duration of each stack with an interval of time, it is possible to use the rich theory that exists in interval graphs in order to create a model based on the completion of a graph with edges. The structure of this type of graphs admits a linear ordering of the vertices that de nes an ordering of the stacks, and consequently decides a sequence for the cutting patterns. The polytope de ned by this formulation is full-dimensional and the main inequalities in the model are proved to be facets. Additional inequalities are derived based on the properties of chordal graphs and comparability graphs. The maximum number of open stacks is related with the chromatic number of the solution graph; thus the formulation is strengthened by adding the representatives formulation for the vertex coloring problem. The model is applied to the minimization of open stacks, and also to the minimum interval graph completion problem and other pattern sequencing problems such as the minimization of the order spread (MORP) and the minimization of the number of tool switches (MTSP). Computational tests of the model are presented.Nesta tese e abordado um problema de optimização que surge em operações de corte de stock chamado minimização do número máximo de pilhas abertas (MOSP) e e proposta uma nova formulação de programação inteira. Associando a duração de cada pilha a um intervalo de tempo, e possível usar a teoria rica que existe em grafos de intervalos para criar um modelo baseado no completamento de um grafo por arcos. A estrutura deste tipo de grafos admite uma ordenação linear dos vértices que define uma ordenação linear das pilhas e, por sua vez, determina a sequência dos padrões de corte. O politopo definido por esta formulação tem dimensão completa e prova-se que as principais desigualdades do modelo são facetas. São derivadas desigualdades adicionais baseadas nas propriedades de grafos cordais e de grafos de comparabilidades. O número máximo de pilhas abertas está relacionado com o número cromático do grafo solução, pelo que o modelo e reforçado com a formulação por representativos para o problema de coloração de vértices. O modelo e aplicado a minimização de pilhas abertas, e também ao problema de completamento mínimo de um grafo de intervalos e a outros problemas de sequenciação de padrões, tais como a minimização da dispersão de encomendas (MORP) e a minimização do número de trocas de ferramentas (MTSP). São apresentados testes computacionais do modelo.Fundação para a Ciência e a Tecnologia (FCT), programa de financiamento QREN-POPH-Tipologia 4.1-Formação Avançada comparticipado pelo Fundo Social Europeu e por fundos do MCTES (Bolsa individual com a refer^encia SFRH/BD/32151/2006) entre 2006 e 2009, e pela Escola Superior de Estudos Industriais e de Gest~ao do Instituto Polit ecnico do Porto (Bolsa PROTEC com a refer^encia SFRH/BD/49914/2009) entre 2009 e 2010

Diameter and Coherence of Monotone Path Graph

University of Minnesota Ph.D. dissertation. May 2015. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); ix, 93 pages.A Zonotope $Z$ is the linear projection of an $n$-cube into $\mathbb{R}^d$. Given a generic linear function $f$, an $f$-monotone path on $Z$ is a path along edges from the $f$-minimizing vertex $-z$ to its opposite vertex $z$. The monotone paths of $Z$ are the vertices of the monotone path graph in which two $f$-monotone paths are adjacent when they differ in a face of $Z$. In our illustration the two red paths are adjacent in the monotone path graph because they differ in the highlighted face. An $f$-monotone path is coherent if it lies on the boundary of a polygon obtained by projecting $Z$ to 2 dimensions. The dotted, thick, red path in Figure 0.1 (see pdf) is coherent because it lies on the boundary after projecting $Z$ to the page. However, there is no equivalent projection for the blue double path. The alternate red path may be coherent or incoherent based on the choice of $f$. The coherent $f$-monotone paths of $Z$ are a set of geometrically distinguished galleries of the monotone path graph. Classifying when incoherent $f$-monotone paths exist is the central question of this thesis. We provide a complete classification of all monotone path graphs in corank 1 and 2, finding all families in which every $f$-monotone path is coherent and showing that all other zonotopes contain at least one incoherent $f$-monotone path. For arrangements of corank 1, we prove that the monotone path graph has diameter equal to the lower bound suggested by Reiner and Roichman using methods of $L_2$-accessibility and illustrate that $L_2$ methods cannot work in corank 2 by finding a monotone path graph which has no $L_2$-accessible nodes. We provide examples to illustrate the monotone path graph and obtain a variety of computational results, of which some are new while others confirm results obtained through different methods. Our primary methods use duality to reformulate coherence as a system of linear inequalities. We classify monotone path graphs using single element liftings and extensions, proving for when $Z$ has incoherent $f$-monotone paths, then any lifting or extension of $Z$ has incoherent $f$-monotone paths too. We complete our classification by finding all monotone path graphs with only coherent $f$-monotone paths and finding a set of minimal obstructions which always have incoherent $f$-monotone paths

Exact Integer Programming Approaches to Sequential Instruction Scheduling and Offset Assignment

The dissertation at hand presents the main concepts and results derived when studying the optimal solution of two NP-hard compiler optimization problems, namely instruction scheduling and offset assignment, by means of integer programming. It is the outcome of several years of research as an assistant at Michael Jünger's computer science chair in Cologne, with the particular aim to apply exact mathematical optimization techniques to real-world problems arising in the domain of technical computer science. The two problems studied are rather unrelated apart from the fact that they both take place during the machine code generation phase of a compiler and deal with the handling of limited resources. Instruction scheduling is about the assignment of issue clock cycles to instructions in the presence of precedence, latency, and resource constraints such that the total time needed to execute all the instructions is minimized. Offset assignment deals with storage layouts of program variables and the efficient use of address registers for accesses to these variables. The objective is to employ specialized instructions in order to minimize the overhead caused by address computations. While instruction scheduling needs to be carried out by almost every present compiler irrespective of the processor architecture, the offset assignment problem occurs mainly in compilers for highly specialized processor designs. Instruction scheduling is a well-studied field where several exact and heuristic approaches have been developed and experimentally evaluated in the past. In this thesis, we concentrate on the basic-block instruction scheduling problem for single-issue processors. Basic blocks are program fragments with no side-entrances and -exits, i.e., every instruction of a basic block needs to be executed before the control flow may leave it and enter another basic block. Single-issue processors are capable of starting the execution of exactly one instruction per clock cycle. A number of techniques to preprocess instances of the basic-block instruction scheduling problem were proposed in the literature and are, with emphasis on the more recent ones that arose since the year 2000, thoroughly reviewed in this thesis. They finally led to a constraint programming approach in 2006 that was shown to solve about 350,000 instances to optimality and where some of these instances comprised up to about 2,500 instructions. The last attempt to tackle the problem using integer programming however dates to a time prior to the publication of the latest preprocessing advances. While being successful on a set of instances that impose very restrictive latency constraints, it was shown to be unable to solve hundreds of instances from the aforementioned benchmark set that comprises also large and varying latencies. In addition, the previous integer programming models were almost all based on so-called time-indexed formulations where decision variables model an explicit assignment of instructions to clock cycles. In this thesis, a completely different and novel approach is taken based on the linear ordering problem, a well-studied combinatorial optimization problem. The new models lead to alternative characterizations of the feasible solutions to the basic-block instruction scheduling problem. These facilitate the employment of advanced integer programming methodologies, in particular the design of branch-and-cut algorithms that can handle larger instances. The formulations are further extended by additional inequalities that can be used as cutting planes. Combined with the preprocessing routines that are partially extended and improved as well, the respective solver implementation eventually turned out to be competitive to the constraint programming method. Reaching this point has taken some years and this thesis presents not only the derived models but also several ideas and byproducts that arose in the meantime, and that can help and inspire researchers even if they aim at the application of different solution methodologies. The starting point regarding the offset assignment problem was a different one because especially exact solution approaches were rather rare prior to the models presented in this thesis. The offset assignment problem arose in the 1990s and is considered in several variants that are of theoretical and practical interest. In the simplest one, a processor is assumed to provide only a single address register and only very restricted possibilities to avoid address computation overhead. However, even this simplest variant, that may serve as a building block for the more complex ones, is already NP-hard and has been studied mainly from a heuristic point of view. The few existing exact solution approaches were not capable to solve moderately sized instances so that the quality of heuristic solutions relative to the optimum was hardly known at all. Again, the inspection of the combinatorial structure of the various problem variants turned out to be the key for designing branch-and-cut implementations that can profit from knowledge about related combinatorial optimization problems. The implementation targeting the simple problem variant was the first capable to optimally solve the majority of about 3,000 instances collected in a standard benchmark set. The method could then be further generalized in two steps. First, in a collaboration with Roberto Castañeda Lozano, additional techniques could be incorporated into the approach in order to handle multiple address registers. Fortunately, the methods could then even be further extended to as well deal with more flexible addressing capabilities. In this way, the thesis at hand does not only answer the question how large the address computation overhead can be when using heuristics, but as well presents first results that allow to analyze the impact of the mentioned increased addressing capabilities on the runtime performance and size of real-world programs

Computing crossing numbers

The graph theoretic problem of crossing numbers has been around for over 60 years, but still very little is known about this simple, yet intricate nonplanarity measure. The question is easy to state: Given a graph, draw it in the plane with the minimum number of edge crossings. A lot of research has been devoted to giving an answer to this question, not only by graph theoreticians, but also by computer scientists. The crossing number is central to areas like chip design and automatic graph drawing. While there are algorithms to solve the problem heuristically, we know that it is in general NP-complete. Furthermore, we do not know if the problem is efficiently approximable, except for some special cases. In this thesis, we tackle the problem using Mathematical Programming. We show how to formulate the crossing number problem as systems of linear inequalities, and discuss how to solve these formulations for reasonably sized graphs to provable optimality in acceptable time--despite its theoretical complexity class. We present non-standard branch-and-cut-and-price techniques to achieve this goal, and introduce an efficient preprocessing algorithm, also valid for other traditional non-planarity measures. We discuss extensions of these ideas to related crossing number variants arising in practice, and show a practical application of a formerly purely theoretic crossing number derivative. The thesis also contains an extensive experimental study of the formulations and algorithms presented herein, and an outlook on its applicability for graph theoretic questions regarding the crossing numbers of special graph classes.Das Kreuzungszahlproblem wird von Graphentheoretikern seit über 60 Jahren betrachtet, jedoch ist noch immer sehr wenig über dieses einfache und zugleich hochkomplizierte Ma der Nichtplanarität bekannt. Die Aufgabenstellung ist simpel: Gegeben ein Graph, zeichnen Sie ihn mit der kleinstmöglichen Anzahl an Kantenkreuzungen. Nicht nur Graphentheoretiker sondern auch Informatiker beschäftigten sich ausgiebig mit dieser Aufgabe, denn es handelt sich dabei um ein zentrales Konzept im Chipdesign und im automatischen Graphenzeichnen. Zwar existieren Algorithmen um das Problem heuristisch zu lösen, jedoch wissen wir, dass es im Allgemeinen NP-vollständig ist. Darüberhinaus ist auch unbekannt, ob sich das Problem, außer in Spezialfällen, effizient approximieren lässt. In dieser Dissertation, versuchen wir das Problem mit Hilfe der Mathematischen Programmierung zu lösen. Wir zeigen, wie man das Kreuzungszahlproblem als verschiedene Systeme von linearen Ungleichungen formulieren kann und diskutieren wie man diese Formulierungen für nicht allzu große Graphen beweisbar optimal und in akzeptabler Zeit lösen kann - unabhängig von seiner formalen Komplexitätsklasse. Wir stellen dazu benötigte maßgeschneiderte Branch-and-Cut-and-Price Techniken vor, und präsentieren einen effizienten Algorithmus zur Vorverarbeitung; dieser ist auch für andere traditionelle Ma e der Nichtplanarität geeignet. Wir diskutieren Erweiterungen unserer Ideen für verwandte Kreuzungszahlkonzepte die in der Praxis auftreten, und zeigen eine praktische Anwendung eines vormals rein theoretisch behandelten Kreuzungszahl-Derivats auf. Diese Arbeit enthält auch eine ausführliche experimentelle Studie der präsentierten Formulierungen und Algorithmen, sowie einen Ausblick über deren mögliche Nutzung für graphentheoretische Fragen bezüglich der Kreuzungszahlen von speziellen Graphenklassen

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum