Kernelization of generic problems : upper and lower bounds

This thesis addresses the kernelization properties of generic problems, defined via syntactical restrictions or by a problem framework. Polynomial kernelization is a formalization of data reduction, aimed at combinatorially hard problems, which allows a rigorous study of this important and fundamental concept. The thesis is organized into two main parts. In the first part we prove that all problems from two syntactically defined classes of constant-factor approximable problems admit polynomial kernelizations. The problems must be expressible via optimization over first-order formulas with restricted quantification; when relaxing these restrictions we find problems that do not admit polynomial kernelizations. Next, we consider edge modification problems, and we show that they do not generally admit polynomial kernelizations. In the second part we consider three types of Boolean constraint satisfaction problems.We completely characterize whether these problems admit polynomial kernelizations, i.e.,given such a problem our results either provide a polynomial kernelization, or they show that the problem does not admit a polynomial kernelization. These dichotomies are characterized by properties of the permitted constraints.Diese Dissertation beschäftigt sich mit der Kernelisierbarkeit von generischen Problemen, definiert durch syntaktische Beschränkungen oder als Problemsystem. Polynomielle Kernelisierung ist eine Formalisierung des Konzepts der Datenreduktion für kombinatorisch schwierige Probleme. Sie erlaubt eine grüdliche Untersuchung dieses wichtigen und fundamentalen Begriffs. Die Dissertation gliedert sich in zwei Hauptteile. Im ersten Teil beweisen wir, dass alle Probleme aus zwei syntaktischen Teilklassen der Menge aller konstantfaktor-approximierbaren Probleme polynomielle Kernelisierungen haben. Die Probleme müssen durch Optimierung über Formeln in Prädikatenlogik erster Stufe mit beschränkter Quantifizierung beschreibbar sein. Eine Relaxierung dieser Beschränkungen gestattet bereits Probleme, die keine polynomielle Kernelisierung erlauben. Im Anschluss betrachten wir Kantenmodifizierungsprobleme und zeigen, dass diese im Allgemeinen keine polynomielle Kernelisierung haben. Im zweiten Teil betrachten wir drei Arten von booleschen Constraint-Satisfaction-Problemen. Wir charakterisieren vollständig welche dieser Probleme polynomielle Kernelisierungen erlauben. Für jedes gegebene Problem zeigen unsere Resultate entweder eine polynomielle Kernelisierung oder sie zeigen, dass das Problem keine polynomielle Kernelisierung hat. Die Dichotomien sind durch Eigenschaften der erlaubten Constraints charakterisiert

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT(Gamma) (respectively CSP(Gamma)) with constraint language F, we mean a value c(0) &amp;gt; 1 such that the problem cannot be solved in time O(c(n)) for any c &amp;lt; c(0) unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O(c(n)) for some c &amp;lt; 2. Such lower bounds have proven extremely elusive, and except for cases where c(0) = 2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages r in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations. Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.Funding Agencies|Swedish resourch council (VR) [2019-03690]</p

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Quantum Algorithms for Scientific Computing and Approximate Optimization

Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations

Problèmes de graphes motivés par des modèles basse et haute résolution de grands assemblages de protéines

To explain the biological function of a molecular assembly (MA), one has to know its structural description. It may be ascribed to two levels of resolution: low resolution (i.e. molecular interactions) and high resolution (i.e. relative position and orientation of each molecular subunit, called conformation). Our thesis aims to address the two problems from graph aspects.The first part focuses on low resolution problem. Assume that the composition (complexes) of a MA is known, we want to determine all interactions ofsubunits in the MA which satisfies some property. It can be modeled as a graph problem by representing a subunit as a vertex, then a subunit interaction is an edge, and a complex is an induced subgraph. In our work, we use the fact that a subunit has a bounded number of interactions. It leads to overlaying graph with bounded maximum degree. For a graph family F and a fixed integer k, given a hypergraph H = (V (H), E(H)) (whose edges are subsets of vertices) and an integer s, M AX (∆ ≤ k)-F -O VERLAY consists in deciding whether there exists a graph with degree at most k such that there are at least s hyperedges in which the subgraph induced by each hyperedge (complex) contains an element of F. When s = |E(H)|, it is called (∆ ≤ k)-F -O VERLAY . We present complexity dichotomy results (P vs. NP-complete) for MAX (∆ ≤ k)-F-OVERLAY and (∆ ≤ k)-F-OVERLAY depending on pairs (F, k).The second part presents our works motivated by high resolution problem. Assume that we are given a graph representing the interactions of subunits, a finite set of conformations for each subunit and a weight function assessing the quality of the contact between two subunits positioned in the assembly. Discrete Optimization of Multiple INteracting Objects (D OMINO ) aims to find conformations for the subunits maximizing a global utility function. We propose a new approach based on this problem in which the weight function is relaxed, CONFLICT COLORING . We present studies from both theoretical and experimental points of view. Regarding the theory, we provide a complexity dichotomy result and also algorithmic methods (approximation and fixed paramater tracktability). Regarding the experiments, we build instances of CONFLICT COLORING associated with Voronoi diagrams in the plane. The obtained statistics provide information on the dependencies of the existences of a solution, to parameters used in ourexperimental setup.Pour comprendre les fonctions biologiques d’un assemblage moléculaire (AM), il est utile d’en avoir une représentation structurale. Celle-ci peut avoir deux niveaux de résolution : basse résolution (i.e. interactions moléculaires) et haute résolution (i.e. position relative et orientation de chaque sous-unité, appelée conformation). Cette thèse s’intéresse à trouver de telles représentations à l’aide de graphes.Dans la première partie, nous cherchons des représentations basse résolution. Etant donné la composition des complexes d’un AM, notre but est de déterminer les interactions entre ses différentes sous-unités. Nous modélisons l’AM à l’aide d’un graphe : les sous-unités sont les sommets, les interactions entre elles sont les arêtes et un complexe est un sous-graphe induit. Utilisant le fait qu’une sous-unité n’a qu’un nombre limité d’interactions, nous arrivons au problème suivant. Pour un graphe F et un entier k fixés, étant donné un hypergraphe H et un entier s, MAX (∆ ≤ k)-F-OVERLAY consiste à décider s’il existe un graphe de degré au plus k tel qu’au moins s hyperarêtes de H induisent un sous-graphe contenant F (en tant que sous-graphe). La restriction au cas s = |E(H)| est appelée (∆ ≤ k)-F-OVERLAY . Nous donnons une dichotomie de complexité (P vs. NP-complet) pour MAX (∆ ≤ k)-F-OVERLAY et (∆ ≤ k)-F-OVERLAY en fonction du couple (F, k).Dans la seconde partie, nous nous attaquons à la haute résolution. Nous sont donnés un graphe représentant les interactions entre sous-unités, un ensemble de conformations possibles pour chaque sous-unité et une fonction de poids représentant la qualité de contact entre les conformations de deux sous-unités interagissant dans l’assemblage. Le problème Discrete Optimization of Multiple INteracting Objects (D OMINO ) consiste alors à trouver les conformations pour les sous-unités qui maximise une fonction d’utilité globale. Nous proposons une nouvelle approche à ce problème en relâchant la fonction de poids, ce qui mène au problème de graphe CONFLICT COLORING . Nous donnons tout d’abord des résultats de complexité et des algorithmes (d’approximation et à paramètre fixé). Nous menons ensuite des expérimentations sur des instances de CONFLICT COLORING associées à des diagrammes de Voronoi dans le plan. Les statistiques obtenues nous informent sur comment les parmètres de notre montage expérimental influe sur l’existence d’une solution

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum