Based on the author's Ph. D. thesis, University of California, Berkeley, 1993. / Includes bibliographical references (p. [73]-78) and index. Also issued online.

Title: Computational Complexity in Graph Theory Author: Eva Jelínková Department: Department of Applied Mathematics Supervisor: Prof. RNDr. Jan Kratochvíl, CSc., Department of Applied Mathematics Abstract: We address problems from graph theory, especially from the computational complexity point of view. In the first part of the thesis we address the computational complexity of problems related to Seidel's switch- ing of graphs. We prove that the problem to decide if a given graph can be switched to contain at most a given number of edges is NP-complete, even for graphs with bounded density. We thus partially answer a question of Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014]. We also describe infinitely many graphs H such that it is NP-complete to decide if a given graph is switching-equivalent to a graph that does not contain H as an induced subgraph. We thus close an open problem of Kratochvíl, Nešetřil and Zýka [Annals of Discrete Math. 51, 1992]. In the second part of the thesis we address the topic of matchings under preferences. We focus on the housing market problem, in particular, on the model with duplicate houses. We present a 2-approximation algorithm for the maximum number of satisfied agents when the preference lists of agents are trichotomic. On the other hand, we...

Nesta tese, nós estudamos a classe de problemas de blocos-móveis. Um problema de blocos-móveis consiste em k blocos móveis dispostos em um labirinto em grade quadrangular onde há um bloco móvel adicional chamado de o homem, que é o único bloco que pode ser movido diretamente. Em particular, cada problema de blocos-móveis é definido pelo conjunto de movimentos disponíveis, pela descrição do objetivo e pelo o que acontece quando o homem tenta mover um bloco. Sokoban é o problema de blocos-móveis mais conhecido e pesquisado. Nós investigamos a complexidade computacional de problemas de blocos-móveis. Antes desta tese, a maior parte da literatura cientifica abordou problemas de blocos-móveis apenas com movimentos de EMPURRAR, na maioria dos casos provando que esses problemas são PSPACE-complete. Nós consideramos dois conjuntos de problemas: apenas movimentos de PUXAR, e movimentos de EMPURRAR e PUXAR combinados. Nossas reduções usam a Lógica de Restrições Não Determinística. Nós provamos que muitos problemas apenas com movimentos de PUXAR são PSPACE-complete. Além disso, nós provamos que o conjunto de problemas com movimentos de EMPURRAR e PUXAR é PSPACE-complete. A nossa contribuição nessa linha de pesquisa é aprimorar o conhecimento sobre o panorama da complexidade de problemas de blocos-móveis. Nosso principal objetivo com essa tese é resolver otimamente problemas de blocos-móveis com foco em Sokoban. Métodos baseados em busca heurística e heurísticas de abstrações como banco de dados de padrões são as abordagens mais efetivas para resolver otimamente esses problemas. Nós fazemos muitas contribuições nessa linha de pesquisa. Nós introduzimos novas funções heurísticas usando bancos de dados padrão com a ideia de estados objetivos intermediários. Propomos uma técnica baseada em bancos de dados padrão para detectar impasses. Propomos regras de desempate que exploram a estrutura do problema. Usando estas funções heurísticas e regras de desempate nós aumentamos o número de instâncias resolvidas de forma ótima de Sokoban e outros problemas em comparação com os métodos anteriores. / In this thesis, we study the class of moving-blocks problems. A moving-blocks problem consists of k movable blocks placed on a grid-square maze where there is an additional movable block called the man, which is the only block that can be moved directly. In particular, each moving-blocks problem is defined by the set of moves available, by the goal description and by what happens when the man attempts to move a block. Sokoban is the best known and researched moving-blocks problem. We study moving-blocks problems in theory and practice. We investigate the computational complexity of problems of moving-blocks. Prior to this thesis, most of the scientific literature addressed moving-blocks problems with PUSH moves only, in most of the cases proving that these problems are PSPACE-complete. We consider two sets of problems: PULL moves only, and PUSH and PULL moves combined. Our reductions are from Nondeterministic Constraint Logic. We prove that many problems with PULL moves only are PSPACE-complete. In addition, we prove that the entire set of PUSH and PULL moves is PSPACE-complete. Our contribution in this research line is to enhance the knowledge on the complexity landscape of moving-blocks problems. Our main objective in this thesis is to optimally solve moving-blocks problems with a focus on Sokoban. Methods based on heuristic search and abstraction heuristics such as pattern databases are the most effective approaches to optimally solve these problems. We make many contributions in this research line. We introduce novel heuristic functions using pattern databases with the idea of intermediate goal states. We propose a technique based on pattern databases to detect deadlocks. We propose tie-breaking rules that exploit the structure of the problem. Using these heuristic functions and tie-breaking rules we increase the number of optimally solved instances of Sokoban and other problems compared to previous methods.

In video transmission there is a need to compose a wide-band signal from a numberof narrow-band sub-signals. A flexible solution offers the possibility to place any narrow-band sub-signal anywhere in the wide-band signal, making better use of the frequency space of the wide-band signal. A multi-standard supportive solution will also consider the three standard bandwidths of digital and analog video transmissions, both terrestrial and cable (6; 7 and 8 MHz), in use today. This thesis work will study the efficiency of a flexible aggregation solution, in terms of computational complexity and error vector magnitude (EVM). The solution uses oversampled complex modulated filter banks and inner channelizers, to reduce the total workload on the system. Each sub-signal is channelized through an analysis filter bank and together all channelized sub-signals are aggregated through one synthesis filter bank to form the wide-band composite signal. The EVM between transmitted and received sub-signals are investigated for an increasing number of sub-signals. The solution in this thesis work is performing good for the tested number of up to 100 narrow-band sub-signals. The result indicates that the multi-standard flexible aggregation solution is efficient for an increasing number of transmitted sub-signals.

L'incertitude a été toujours présente dans les problèmes d'optimisation. Dans ce travail, nous nous intéressons aux problèmes d'optimisation multi-niveaux où l'incertitude apparaît très naturellement. Les problèmes d'optimisation multi-niveaux avec incertitude ont suscité un intérêt à la fois théorique et pratique. L'optimisation robuste fait partie des méthodes les plus étudiées pour traiter ces problèmes. En optimisation robuste, nous cherchons une solution qui optimise la fonction objective pour le pire scénario appartenant à un ensemble d'incertitude donné. Les problèmes d'optimisation robuste multi-niveaux sont difficiles à résoudre, même de façon heuristique. Dans cette thèse, nous abordons les problèmes d'optimisation robuste à travers le prisme des méthodes de décomposition. Ces méthodes décomposent le problème en un problème maître (MP) et plusieurs problèmes satellites de séparation (AP). Dans ce contexte, les solutions et les relaxations heuristiques ont une importance particulière. Même pour les problèmes d'optimisation combinatoires, les relaxations sont importantes pour analyser l'écart de l'optimalité des solutions heuristiques. Un autre aspect important est l'utilisation des heuristiques comme integrés dans une méthode exacte. Les principales contributions de ce travail sont les suivantes. Premièrement, nous proposons une nouvelle relaxation pour les problèmes multi-niveaux basée sur l’approche dite d’information parfaite dans le domaine de l’optimisation stochastique. L'idée principale derrière cette méthode est d'éliminer les contraintes de non anticipativité du modèle pour obtenir un problème plus simple. Nous pouvons ensuite fournir des algorithmes combinatoires ad-hoc et des formulations de programmation mixte en nombres entiers compactes pour ce problème. Deuxièmement, nous proposons de nouveaux algorithmes de programmation dynamique pour résoudre les problèmes satellites apparaissant dans une classe spécifique de problèmes robustes pour un ensemble d'incertitude de type budget. Ce type d'incertitude est basé sur le nombre maximum d'écarts autorisés et leur taille. Ces algorithmes peuvent être appliqués à des problèmes de lot-sizing et à des problèmes de tournées de véhicules. Enfin, nous proposons un modèle robuste pour un problème lié à l’installation équitable de capteurs. Ce modèle fait le lien entre l'optimisation robuste et l'optimisation stochastique avec contraintes probabilistes ambigües. / Uncertainty has always been present in optimization problems, and it arises even more severely in multistage optimization problems. Multistage optimization problems underuncertainty have attracted interest from both the theoretical and the practical level.Robust optimization stands among the most established methodologies for dealing with such problems. In robust optimization, we look for a solution that optimizes the objective function for the worst possible scenario, in a given uncertainty set. Robust multi-stage optimization problems are hard to solve even heuristically. In this thesis, we address robust optimization problems through the lens of decompositions methods. These methods are based on the decomposition of the robust problem into a master problem (MP) and several adversarial separation problems (APs). The master problem contains the original robust constraints, however, written only for finite numbers of scenarios. Additional scenarios are generated on the y by solving the APs. In this context, heuristic solutions and relaxations have a particular importance. Similarly to combinatorial optimization problems, relaxations are important to analyze the optimality gap of heuristic solutions. Heuristic solutions represent a substantial gain from the computational viewpoint, especially when used to solve the separation problem. Because the adversarial problems must be solved several times, good heuristic solution may avoid the exact solution of the APs. The main contributions of this work are three-fold. First, we propose a new relaxation for multi-stage problems based on the approach named perfect information in the field of stochastic optimization. The main idea behind this method is to remove nonanticipativity constraints from the model to obtain a simpler problem for which we can provide ad-hoc combinatorial algorithms and compact mixed integer programming formulations. Second, we propose new dynamic programming algorithms to solve the APs for robust problems involving budgeted uncertainty, which are based on the maximum number of deviations allowed and on the size of the deviations. These algorithms can be applied to lot-sizing problems and vehicle routing problems among others. Finally, we study the robust equitable sensor location problem. We make the connection between the robust optimization and the stochastic programming with ambiguous probabilistic constraints. We propose linear models for several variants of the problem together withnumerical results.

Nesta tese, nós estudamos a classe de problemas de blocos-móveis. Um problema de blocos-móveis consiste em k blocos móveis dispostos em um labirinto em grade quadrangular onde há um bloco móvel adicional chamado de o homem, que é o único bloco que pode ser movido diretamente. Em particular, cada problema de blocos-móveis é definido pelo conjunto de movimentos disponíveis, pela descrição do objetivo e pelo o que acontece quando o homem tenta mover um bloco. Sokoban é o problema de blocos-móveis mais conhecido e pesquisado. Nós investigamos a complexidade computacional de problemas de blocos-móveis. Antes desta tese, a maior parte da literatura cientifica abordou problemas de blocos-móveis apenas com movimentos de EMPURRAR, na maioria dos casos provando que esses problemas são PSPACE-complete. Nós consideramos dois conjuntos de problemas: apenas movimentos de PUXAR, e movimentos de EMPURRAR e PUXAR combinados. Nossas reduções usam a Lógica de Restrições Não Determinística. Nós provamos que muitos problemas apenas com movimentos de PUXAR são PSPACE-complete. Além disso, nós provamos que o conjunto de problemas com movimentos de EMPURRAR e PUXAR é PSPACE-complete. A nossa contribuição nessa linha de pesquisa é aprimorar o conhecimento sobre o panorama da complexidade de problemas de blocos-móveis. Nosso principal objetivo com essa tese é resolver otimamente problemas de blocos-móveis com foco em Sokoban. Métodos baseados em busca heurística e heurísticas de abstrações como banco de dados de padrões são as abordagens mais efetivas para resolver otimamente esses problemas. Nós fazemos muitas contribuições nessa linha de pesquisa. Nós introduzimos novas funções heurísticas usando bancos de dados padrão com a ideia de estados objetivos intermediários. Propomos uma técnica baseada em bancos de dados padrão para detectar impasses. Propomos regras de desempate que exploram a estrutura do problema. Usando estas funções heurísticas e regras de desempate nós aumentamos o número de instâncias resolvidas de forma ótima de Sokoban e outros problemas em comparação com os métodos anteriores. / In this thesis, we study the class of moving-blocks problems. A moving-blocks problem consists of k movable blocks placed on a grid-square maze where there is an additional movable block called the man, which is the only block that can be moved directly. In particular, each moving-blocks problem is defined by the set of moves available, by the goal description and by what happens when the man attempts to move a block. Sokoban is the best known and researched moving-blocks problem. We study moving-blocks problems in theory and practice. We investigate the computational complexity of problems of moving-blocks. Prior to this thesis, most of the scientific literature addressed moving-blocks problems with PUSH moves only, in most of the cases proving that these problems are PSPACE-complete. We consider two sets of problems: PULL moves only, and PUSH and PULL moves combined. Our reductions are from Nondeterministic Constraint Logic. We prove that many problems with PULL moves only are PSPACE-complete. In addition, we prove that the entire set of PUSH and PULL moves is PSPACE-complete. Our contribution in this research line is to enhance the knowledge on the complexity landscape of moving-blocks problems. Our main objective in this thesis is to optimally solve moving-blocks problems with a focus on Sokoban. Methods based on heuristic search and abstraction heuristics such as pattern databases are the most effective approaches to optimally solve these problems. We make many contributions in this research line. We introduce novel heuristic functions using pattern databases with the idea of intermediate goal states. We propose a technique based on pattern databases to detect deadlocks. We propose tie-breaking rules that exploit the structure of the problem. Using these heuristic functions and tie-breaking rules we increase the number of optimally solved instances of Sokoban and other problems compared to previous methods.

La complexité implicite (ICC) vise à donner des caractérisations de classes de complexité dans des langages de programmation ou des logiques, sans faire référence à des bornes sur les ressources (temps, espace mémoire). Dans cette thèse, nous étudions l’approche de la logique linéaire à la complexité implicite. L’objectif est de donner des caractérisations de classes de complexité, à travers des variantes du lambda-calcul qui sont typables dans de tels systèmes. En particulier, nous considérons à la fois une perspective monovalente et une perspective polyvalente par rapport à l’ICC. Dans le premier cas, le but est de caractériser une hiérarchie de classes de complexité à travers un lambda-calcul élémentaire typé dans la logique linéaire élémentaire (ELL), où la complexité ne dépend que de l’interface d’un programme, c’est à dire son type. La deuxième approche rend compte à la fois des fonctions calculables en temps polynomial et de la normalisation forte, à travers des termes du lambda-calcul pur qui sont typés dans un système inspiré par la logique linéaire Soft (SLL); en particulier, par rapport à l’approche logique ordinaire, ici nous abandonnons la modalité “!” en faveur de l’emploi des types stratifiés, vus comme un raffinement des types intersection non associatifs, afin d’améliorer la typabilité et, en conséquence, l’expressivité. Enfin, nous explorons l’utilisation des types intersection, privés de certaines de leurs propriétés, vers une direction plus quantitative que l’approche qualitative habituelle, afin d’obtenir une borne sur le calcul de lambda-termes purs, en obtenant en plus une caractérisation de la normalisation forte. / In this thesis we explore the linear logic approach to implicit computational complexity, through the design of type assignment systems based on light linear logic, or heavily inspired by them, with the purpose of giving a characterization of one or more complexity classes, through variants of lambda-calculi which are typable in such systems. In particular, we consider both a monovalent and a polyvalent perspective with respect to ICC. In the first one the aim is to characterize a hierarchy of complexity classes through an elementary lambda-calculus typed in Elementary Linear Logic (ELL), where the complexity depends only on the interface of a term, namely its type. The second approach gives an account of both the functions computable in polynomial time and of strong normalization, through terms of pure lambda-calculus which are typed in a system inspired by Soft Linear Logic (SLL); in particular, with respect to the usual logical take, in the latter we give up the “!” modality in favor of employing stratified types as a refinement of non-associative intersection types, in order to improve typability and, as a consequence, expressivity.Finally we explore the use of intersection types, deprived of some of their usual properties, towards a more quantitative approach rather than the usual qualitative one, namely in order to compute a bound on the computation of pure lambda-terms, obtaining in addition a characterization of strong normalization.

Using concepts of the socio-technical systems theory, this dissertation defines mathematically the problems of cooperative teams formation considering social and technical constraints separately, and then presents their computational complexity. Mainly, it is defined and studied the central problem in this work, which jointly considers social and technical requirements for creating teams of cooperative work, to be called FEST (Socio-Technical Teams Formation Problem). Two mathematical formulations and a meta-heuristic are proposed for FEST. One formulation uses a cubic number of variables and constraints, whereas the second one has a quadratic number of variables but an exponential number of constraints. The proposed heuristic is based on the Non-monotonic Simulated Annealing meta-heuristic with local search using swap-like operators. The correctness of both formulations is proved. A polynomial algorithm to separate the constraints of the second formulation is presented. It is proved that the two formulations provide the same linear programming bound, and valid inequalities to strengthen it are proposed. For the compact formulation, some classes of valid inequalities are shown to be facet-inducing under suitable hypotheses. Finally, it is statistically analyzed the performance of the presented formulations and meta-heuristic. Real and random generated instances are used in the computational experiments. / Utilizando conceitos da Teoria dos Sistemas SociotÃcnicos, este trabalho define matematicamente os problemas de formaÃÃo de equipes cooperativas considerando separadamente restriÃÃes sociais e tÃcnicas e apresenta a complexidade computacional dos mesmos. Sobretudo, à definido e estudado o problema central deste trabalho, que considera conjuntamente requisitos sociais e tÃcnicos para criaÃÃo de equipes de trabalho cooperativo, denominado FEST (Problema de FormaÃÃo de Equipes SociotÃcnicas). Duas formulaÃÃes matemÃticas e uma meta-heurÃstica para o FEST sÃo propostas. Uma formulaÃÃo utiliza um nÃmero cÃbico de variÃveis e restriÃÃes, enquanto a segunda formulaÃÃo possui um nÃmero quadrÃtico de variÃveis, mas um nÃmero exponencial de restriÃÃes. A meta-heurÃstica proposta à baseada no Simulated Annealing NÃo-MonotÃnico com busca local que usa operadores tipo swap. A corretude de ambas as formulaÃÃes à provada. Um algoritmo polinomial para separar as restriÃÃes da segunda formulaÃÃo à apresentado. Mostra-se que as duas formulaÃÃes fornecem o mesmo limite de programaÃÃo linear, e desigualdades vÃlidas para fortalecÃ-lo sÃo propostas. Para a formulaÃÃo compacta, algumas classes de desigualdades vÃlidas sÃo demonstradas indutoras de facetas sob hipÃteses apropriadas. Por fim, foi analisado estatisticamente o desempenho das formulaÃÃes e da meta-heurÃstica apresentadas. InstÃncias reais e geradas aleatoriamente sÃo usadas nos experimentos computacionais.

FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / Introduzimos o problema de Ãrvore Geradora com DependÃncias MÃnima, AGDM(G,D,w), definido sobre um grafo G(V,E) e um digrafo D(E,A), cujos vÃrtices sÃo as arestas de G e cujos arcos definem dependÃncias entre tais arestas. O problema consiste em encontrar, dentre as Ãrvores geradoras do grafo G(V,E) que satisfaÃam as restriÃÃes de dependÃncia impostas pelo digrafo de entrada D(E,A), uma que tenha custo mÃnimo, segundo a ponderaÃÃo w das arestas de G. As restriÃÃes de dependÃncia exigem que uma aresta e de G sà pode fazer parte de uma soluÃÃo se for uma fonte em D ou se fizer parte da soluÃÃo alguma outra aresta à tal que o arco (e′, e) esteja em D. Provamos que decidir se hà soluÃÃo viÃvel para AGDM(G,D,w) à um problema NP-completo, mesmo quando G à um cacto cordal e D à a uniÃo de arborescÃncias de altura no mÃximo 2. Sua NP-completude tambÃm à mostrada ainda que G seja bipartido, as restriÃÃes de dependÃncia ocorram apenas entre arestas adjacentes de G e formem arborescÃncias de altura no mÃximo 2. Resultados idÃnticos sÃo obtidos para as variantes do problema onde, nas restriÃÃes de dependÃncia, substitui-se o requisito âalgumaâ por âexatamente umaâ ou âtodaâ. Para resolver o problema, apresentamos algumas formulaÃÃes de programaÃÃo inteira e desigualdades vÃlidas. Propomos uma estratÃgia para reduzir a dimensÃo do problema, excluindo arestas de G com base na estrutura de D. Avaliamos os modelos e algoritmos propostos usando instÃncias geradas aleatoriamente. Resultados computacionais sÃo reportados. / We introduce the Dependency Constrained Minimum Spanning Tree Problem, DCMST(G,D,w), defined over a graph G(V,E) and a digraph D(E,A), whose vertices are the edges of G and whose arcs describe dependency relations between these edges. Such problem consists of finding, among the spanning trees of G(V,E) satisfying the dependency constraints imposed by D(E,A), that one whose cost is minimum, according to a edgeweight function w. The dependency constraints impose that an edge e of G can be part of a solution either if it is a source in D or if some other edge e′, such that the arc (e′, e) is in D, is part of it as well. We prove that deciding whether there is a feasible solution to DCMST(G,D,w) is an NP-complete problem, even if G is a chordal cactus and D is a union of arborescences of height at most 2. NP-completeness also applies if G is bipartite, the dependency constraints occur only between adjacent edges of G and their related arcs describe arborescences whose height is at most 2. The same results are obtained for the problem variants which demand that, instead of âsomeâ, âexactly oneâor âallâdependencies be part of a solution. To solve the problem, we introduce some integer programming formulations and some valid inequalities. We propose a strategy to reduce the problem dimension by excluding some edges of G according to the structure of D. We evaluate the introduced models and algorithms using randomly generated instances. Computational results are reported.

The theory of Grobner bases has garnered the interests of a large number of researchers in computational algebra due to its applications not only in mathematics but also in areas like control systems, robotics, cryptography to name a few. It is well known that the computation of Grobner bases takes time doubly exponential in the number of indeterminates rendering it impractical in all but a few places.The current known algorithms for Grobner bases depend on the term order over which Grobner bases is computed. In this thesis, we study computational complexity of some problems in computational ideal theory. We also study the algebraic formulation of combinatorial optimization problems. Gritzmann and Sturmfels (1993) posed the following question: Given a set of generators, decide whether it is a Gr¨obner bases with respect to some term order. This problem, termed as the Grobner Basis Detection(GBD)problem, was introduced as an application of Minkowski addition of polytopes. It was shown by Sturmfels and Wiegelmann (1997) that GBD is NP-hard. We study the problem for the case of zero-dimensional ideals and show that the problem is hard even in this special case. We study the detection problem in the case of border bases which are an alternative to Grobner bases in the case of zero dimensional ideals. We propose the Border Basis Detection(BBD) problem which is defined as follows: Given a set of generators of an ideal, decide whether that set of generators is a border basis of the ideal with respect to some order ideal. It is shown that BBD is NP-complete. We also formulate the rainbow connectivity problem as a system of polynomial equations such that solving the polynomial system yields a solution to it. We give an alternate formulation of the rainbow connectivity problem as a membership problem in polynomial ideals.