Synchronous Context-Free Grammars and Optimal Linear Parsing Strategies

Synchronous Context-Free Grammars (SCFGs), also known as syntax-directed translation schemata, are unlike context-free grammars in that they do not have a binary normal form. In general, parsing with SCFGs takes space and time polynomial in the length of the input strings, but with the degree of the polynomial depending on the permutations of the SCFG rules. We consider linear parsing strategies, which add one nonterminal at a time. We show that for a given input permutation, the problems of finding the linear parsing strategy with the minimum space and time complexity are both NP-hard

Exact and Approximate Digraph Bandwidth

In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively. - Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree. - Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor. - Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n). - Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n)

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

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

Engineering combinatorial optimization algorithms to improve the lifetime of OLED displays

In this thesis, we consider an optimization problem arising in the design of controllers for OLED displays. Our objective is to minimize amplitude of the electrical current through the diodes which has a direct impact on the lifetime of such a display. To this end, we model this problem as an integer linear program. Subsequently, we refine this formulation by exploiting the combinatorial structure of the problem. We establish a general approximation framework based on network flow techniques. We apply an algorithm engineering approach to improve our heuristics by interacting experimental evaluations and theoretical investigations. For example on the theoretical side, we present a fully combinatorial linear-time algorithm for the MAXFLOW problem on graphs with bounded bandwidth and accordingly numbered nodes. With respect to our application, we implement a custom-made version of this algorithm to solve an important subproblem. Finally, we develop fully combinatorial approximation algorithms well suited for being implemented in the hardware of a control device that drives an OLED display. Moreover, our algorithms achieve near optimal solutions in practice yielding significantly longer lifetimes of passive matrix OLED displays.In dieser Arbeit betrachten wir ein Optimierungsproblem, das im Zusammenhang mit der Entwicklung von Steuergeräten für OLED Bildschirme auftritt. Unser Ziel ist es, die Amplitude der Stromstärke, die einen direkten Einfluss auf die Lebensdauer eines solchen Bildschirms hat, zu minimieren. Dazu modellieren wir das Problem als ein ganzzahliges lineares Programm. Danach verfeinern wir diese Formulierung, indem wir die kombinatorische Struktur des Problems ausnutzen. Wir erstellen ein allgemeines Approximationssystem basierend auf Netzwerkflusstechniken. Wir wenden einen algorithm engineering Ansatz an, um unsere Heuristiken durch wechselwirkende experimentelle Auswertungen und theoretische Untersuchungen zu verbessern. Zum Beispiel zeigen wir auf der theoretischen Seite einen vollkombinatorischen Linearzeitalgorithmus für das MAXFLOW Problem auf Graphen mit beschränkter Bandbreite und entsprechend nummerierten Knoten. In Bezug auf unsere Anwendung implementieren wir eine maßgeschneiderte Version dieses Algorithmus, um ein wichtiges Teilproblem zu lösen. Letztendlich haben wir einen vollkombinatorischen Approximationsalgorithmus entwickelt, der gut geeignet ist, um als Schaltkreis in ein Steuergerät für einen OLED Bildschirm integriert zu werden. Darüber hinaus erreichen unsere Algorithmen nahezu optimale Lösungen in der Praxis, was zu einer deutlich längeren Lebensdauer eines OLED Bildschirms mit passiver Matrix führt

Parameterization of Tensor Network Contraction

We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable

Using interval graphs in an order processing optimization problem

In this paper we address an order processing optimization problem known as minimization of open stacks (MOSP). We present an integer pro gramming model, based on the existence of a perfect elimination scheme in interval graphs, which finds an optimal sequence for the costumers orders

The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs

AbstractIn this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in “yes”-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W[t] for all t∈N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete