LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Paths and walks, forests and planes : arcadian algorithms and complexity

This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und Komplexitätstheorie. Wir entwickeln eine neue Technik für schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender Graßmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener Länge in gerichteten Graphen schneller als bisher zu schätzen. Außerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener Länge zu finden, falls der Eingabegraph nur wenige solche Pfade enthält. Übrigens entwickeln wir schnellere deterministische Algorithmen, um Spannbäume mit wenigen Blättern zu finden. Wir studieren außerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-Komplexität davon, die genaue Anzahl von Wäldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollständigen wir die Komplexitätsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass Wälder mit gegebener Kantenzahl zu zählen, wenigstens so schwer ist, wie Cliquen gegebener Größe zu zählen.Cluster of Excellence (Multimodal Computing and Interaction

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

View generated database

This document represents the final report for the View Generated Database (VGD) project, NAS7-1066. It documents the work done on the project up to the point at which all project work was terminated due to lack of project funds. The VGD was to provide the capability to accurately represent any real-world object or scene as a computer model. Such models include both an accurate spatial/geometric representation of surfaces of the object or scene, as well as any surface detail present on the object. Applications of such models are numerous, including acquisition and maintenance of work models for tele-autonomous systems, generation of accurate 3-D geometric/photometric models for various 3-D vision systems, and graphical models for realistic rendering of 3-D scenes via computer graphics

Applications of the Adversary Method in Quantum Query Algorithms

In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.Comment: PhD Thesis, 169 page

Advances in combined architecture, plant, and control design

The advancement of many engineering systems relies on novel design methodologies, design formulations, design representations, and other advancements. In this dissertation, we consider three broad design domains: architecture, plant, and control. These domains cover most of the potential design decision elements in an actively-controlled engineering system. In this dissertation, strategic aspects of this combined problem are addressed. The task of representing and generating candidate architectures is addressed with methods developed based on colored graphs built by enumerating all perfect matchings of a specified catalog of components. The proposed approach captures all architectures under specific assumptions. General combined plant and control design (or co-design) problems are examined. Previous work in co-design theory imposed restrictions on the type of problems that could be posed. Here many of those restrictions are lifted. The problem formulations and optimality conditions for both the simultaneous and nested solution strategies are given along with a detailed discussion of the two methods. Direct transcription is also discussed as it enables the solution of general co-design problems by approximating the problem. Motivated primarily by the need for efficient methods to solve certain control problems that emerge using the nested co-design method, an automated problem generation procedure is developed to support easy specification of linear-quadratic dynamic optimization problems using direct transcription and quadratic programming. Pseudospectral and single-step methods (including the zero-order hold) are all implemented in this unified framework and comparisons are made. Three detailed engineering design case studies are presented. The results from the enumeration and evaluation of all passive analog circuits with up to a certain number of components are used to synthesize low-pass filters and circuits that match a certain magnitude response. Advantages and limitations of enumerative approaches are highlighted in this case study, along with comparisons to circuits synthesized via evolutionary computation; many similarities are found in the topologies. The second case study tackles a complex co-design problem with the design of strain-actuated solar arrays for spacecraft precision pointing and jitter reduction. Nested co-design is utilized along with a linear-quadratic inner loop problem to obtain solutions efficiently. A simpler, scaled problem is analyzed to gain general insights into these results. This is accomplished with a unified theory of scaling in dynamic optimization. The final case study involves the design of active vehicle suspensions. All three design domains are considered in this problem. A class of architecture, plant, and control design problems which utilize linear physical elements is discussed. This problem class can be solved using the methods in this dissertation

Lower Bounds on Quantum Query and Learning Graph Complexities

In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learning graph complexity of the functions. Most notably, we prove an almost optimal Omega(n^(9/7)/sqrt(log n)) lower bound on the non-adaptive learning graph complexity of the Triangle problem. We also prove an Omega(n^(1-2^(k-2)/(2^k-1))) lower bound on the adaptive learning graph complexity of the k-Distinctness problem, which matches the complexity of the best known quantum query algorithm for this problem. Second, we construct optimal adversary lower bounds for various decision problems. Our main procedure for constructing them is to embed the adversary matrix into a larger matrix whose properties are easier to analyze. This embedding procedure imposes certain requirements on the size of the input alphabet. We prove optimal Omega(n^(1/3)) adversary lower bounds for the Collision and Set Equality problems, provided that the alphabet size is at least Omega(n^2). An optimal lower bound for Collision was previously proven using the polynomial method, while our lower bound for Set Equality is new. (An optimal lower bound for Set Equality was also independently and at about the same time proven by Zhandry using the polynomial method [arXiv, 2013].) We compare the power of non-adaptive learning graphs and quantum query algorithms that only utilize the knowledge on the possible positions of certificates in the input string. To do that, we introduce a notion of a certificate structure of a decision problem. Using the adversary method and the dual formulation of the learning graph complexity, we show that, for every certificate structure, there exists a decision problem possessing this certificate structure such that its non-adaptive learning graph and quantum query complexities differ by at most a constant multiplicative factor. For a special case of certificate structures, we construct a relatively general class of problems having this property. This construction generalizes the adversary lower bound for the k-Sum problem derived recently by Belovs and Spalek [ACM ITCS, 2013]. We also construct an optimal Omega(n^(2/3)) adversary lower bound for the Element Distinctness problem with minimal non-trivial alphabet size, which equals the length of the input. Due to the strict requirement on the alphabet size, here we cannot use the embedding procedure, and the construction of the adversary matrix heavily relies on the representation theory of the symmetric group. While an optimal lower bound for Element Distinctness using the polynomial method had been proven for any input alphabet, an optimal adversary construction was previously only known for alphabets of size at least Omega(n^2). Finally, we introduce the Enhanced Find-Two problem and we study its query complexity. The Enhanced Find-Two problem is, given n elements such that exactly k of them are marked, find two distinct marked elements using the following resources: (1) one initial copy of the uniform superposition over all marked elements, (2) an oracle that reflects across this superposition, and (3) an oracle that tests if an element is marked. This relational problem arises in the study of quantum proofs of knowledge. We prove that its query complexity is Theta(min{sqrt(n/k),sqrt(k)})

Introduction to the Modeling and Analysis of Complex Systems

Keep up to date on Introduction to Modeling and Analysis of Complex Systems at http://bingweb.binghamton.edu/~sayama/textbook/! Introduction to the Modeling and Analysis of Complex Systems introduces students to mathematical/computational modeling and analysis developed in the emerging interdisciplinary field of Complex Systems Science. Complex systems are systems made of a large number of microscopic components interacting with each other in nontrivial ways. Many real-world systems can be understood as complex systems, where critically important information resides in the relationships between the parts and not necessarily within the parts themselves. This textbook offers an accessible yet technically-oriented introduction to the modeling and analysis of complex systems. The topics covered include: fundamentals of modeling, basics of dynamical systems, discrete-time models, continuous-time models, bifurcations, chaos, cellular automata, continuous field models, static networks, dynamic networks, and agent-based models. Most of these topics are discussed in two chapters, one focusing on computational modeling and the other on mathematical analysis. This unique approach provides a comprehensive view of related concepts and techniques, and allows readers and instructors to flexibly choose relevant materials based on their objectives and needs. Python sample codes are provided for each modeling example. This textbook is available for purchase in both grayscale and color via Amazon.com and CreateSpace.com.https://knightscholar.geneseo.edu/oer-ost/1013/thumbnail.jp