Eigenschaften chromatischer Polynome

Die Berechnung des 1912 von Birkhoff eingeführten chromatischen Polynoms eines Graphen stellt bekanntlich ein NP-vollständiges Problem dar. Dieses gilt somit erst recht für die Verallgemeinerung des chromatischen Polynoms zum bivariaten chromatischen Polynom nach Dohmen, Pönitz und Tittmann aus dem Jahre 2003. Eine von Averbouch, Godlin und Makowsky 2008 vorgestellte Rekursionsformel verursacht durch wiederholte Anwendung im Allgemeinen einen exponentiellen Rechenaufwand. Daher war das Ziel der vorliegenden Dissertation, Vereinfachungen zur Berechnung des bivariaten chromatischen Polynoms spezieller Graphentypen zu finden. Hierbei wurden folgende Resultate erzielt: Für Vereinigungen von Sternen, für vollständige Graphen, aus welchen die Kanten von Sternen mit paarweise voneinander verschiedenen Ecken gelöscht wurden, für spezielle Splitgraphen und für vollständig partite Graphen konnten rekursionsfreie Gleichungen zur Berechnung des bivariaten chromatischen Polynoms mit jeweils linear beschränkter Rechenzeit gefunden werden. Weiterhin werden Möglichkeiten der Reduktion allgemeiner Splitgraphen, bestimmter bipartiter Graphen sowie vollständig partiter Graphen vorgestellt. Bei letzteren erweist sich eine hierbei gefundene Rekursionsformel durch eine polynomiell beschränkte Laufzeit als effektive Methode. Ferner konnte in einem Abschnitt zu Trennern in Graphen gezeigt werden, dass der Spezialfall der trennenden Cliquen, welcher im univariaten Fall sehr einfach ist, im bivariaten Fall sehr komplexe Methoden erfordert. Ein Zusammenhang zwischen dem bivariaten chromatischen Polynom und dem Matchingpolynom wurde für vollständige Graphen, welchen die Kanten von Sternen mit paarweise voneinander verschiedenen Ecken entnommen wurden, sowie für Bicliquen hergestellt. Die vorliegende Dissertation liefert darüber hinaus auch einige Untersuchungen zum trivariaten chromatischen Polynom, welches auf White (2011) zurückgeht und eine weitere Verallgemeinerung des bivariaten chromatischen Polynoms darstellt. Hierbei konnte gezeigt werden, dass dessen Berechnung selbst für einfache Graphentypen schon recht kompliziert ist. Dieses trifft sogar dann noch zu, wenn man die einzelnen Koeffizienten als bivariate Polynome abspaltet und einzeln berechnet. Abschließend stellt die Arbeit zu vielen Resultaten Implementierungen mit dem Computeralgebrasystem Mathematica bereit, welche zahlreiche Möglichkeiten zu eigenständigen Versuchen bieten.

Computational Structure of Human Language

The central thesis of this report is that human language is NP-complete. That is, the process of comprehending and producing utterances is bounded above by the class NP, and below by NP-hardness. This constructive complexity thesis has two empirical consequences. The first is to predict that a linguistic theory outside NP is unnaturally powerful. The second is to predict that a linguistic theory easier than NP-hard is descriptively inadequate. To prove the lower bound, I show that the following three subproblems of language comprehension are all NP-hard: decide whether a given sound is possible sound of a given language; disambiguate a sequence of words; and compute the antecedents of pronouns. The proofs are based directly on the empirical facts of the language user's knowledge, under an appropriate idealization. Therefore, they are invariant across linguistic theories. (For this reason, no knowledge of linguistic theory is needed to understand the proofs, only knowledge of English.) To illustrate the usefulness of the upper bound, I show that two widely-accepted analyses of the language user's knowledge (of syntactic ellipsis and phonological dependencies) lead to complexity outside of NP (PSPACE-hard and Undecidable, respectively). Next, guided by the complexity proofs, I construct alternate linguisitic analyses that are strictly superior on descriptive grounds, as well as being less complex computationally (in NP). The report also presents a new framework for linguistic theorizing, that resolves important puzzles in generative linguistics, and guides the mathematical investigation of human language.

Modeling of carrier transport in nanowires

We consider a physical model of ultrafast evolution of an initial electron distribution in a quantum wire. The electron evolution is described by a quantum-kinetic equation accounting for the interaction with phonons. A Monte Carlo approach has been developed for solving the equation. The corresponding Monte Carlo algorithm is NP-hard problem concerning the evolution time. To obtain solutions for long evolution times with small stochastic error we combine both variance reduction techniques and distributed computations. Grid technologies are implemented due to the large computational efforts imposed by the quantum character of the model.

Mobility based energy efficient and multi-sink algorithms for consumer home networks

With the fast development of the Internet, wireless communications and semiconductor devices, home networking has received significant attention. Consumer products can collect and transmit various types of data in the home environment. Typical consumer sensors are often equipped with tiny, irreplaceable batteries and it therefore of the utmost importance to design energy efficient algorithms to prolong the home network lifetime and reduce devices going to landfill. Sink mobility is an important technique to improve home network performance including energy consumption, lifetime and end-to-end delay. Also, it can largely mitigate the hot spots near the sink node. The selection of optimal moving trajectory for sink node(s) is an NP-hard problem jointly optimizing routing algorithms with the mobile sink moving strategy is a significant and challenging research issue. The influence of multiple static sink nodes on energy consumption under different scale networks is first studied and an Energy-efficient Multi-sink Clustering Algorithm (EMCA) is proposed and tested. Then, the influence of mobile sink velocity, position and number on network performance is studied and a Mobile-sink based Energy-efficient Clustering Algorithm (MECA) is proposed. Simulation results validate the performance of the proposed two algorithms which can be deployed in a consumer home network environment.

Metaheurísticas evolutivas para o problema de roteamento de unidades móveis de pistoneio

This paper presents metaheuristic strategies based on the framework of evolutionary algorithms (Genetic and Memetic) with the addition of Technical Vocabulary Building for solving the Problem of Optimizing the Use of Multiple Mobile Units Recovery of Oil (MRO units). Because it is an NP-hard problem, a mathematical model is formulated for the problem, allowing the construction of test instances that are used to validate the evolutionary metaheuristics developed

Uma implementação paralela híbrida para o problema do caixeiro viajante usando algoritmos genéticos, GRASP e aprendizagem por reforço

The metaheuristics techiniques are known to solve optimization problems classified as NP-complete and are successful in obtaining good quality solutions. They use non-deterministic approaches to generate solutions that are close to the optimal, without the guarantee of finding the global optimum. Motivated by the difficulties in the resolution of these problems, this work proposes the development of parallel hybrid methods using the reinforcement learning, the metaheuristics GRASP and Genetic Algorithms. With the use of these techniques, we aim to contribute to improved efficiency in obtaining efficient solutions. In this case, instead of using the Q-learning algorithm by reinforcement learning, just as a technique for generating the initial solutions of metaheuristics, we use it in a cooperative and competitive approach with the Genetic Algorithm and GRASP, in an parallel implementation. In this context, was possible to verify that the implementations in this study showed satisfactory results, in both strategies, that is, in cooperation and competition between them and the cooperation and competition between groups. In some instances were found the global optimum, in others theses implementations reach close to it. In this sense was an analyze of the performance for this proposed approach was done and it shows a good performance on the requeriments that prove the efficiency and speedup (gain in speed with the parallel processing) of the implementations performed

Estudo Comparativo de Métricas de Pontuação para Aprendizagem Estrutural de Redes Bayesianas

Bayesian networks are powerful tools as they represent probability distributions as graphs. They work with uncertainties of real systems. Since last decade there is a special interest in learning network structures from data. However learning the best network structure is a NP-Hard problem, so many heuristics algorithms to generate network structures from data were created. Many of these algorithms use score metrics to generate the network model. This thesis compare three of most used score metrics. The K-2 algorithm and two pattern benchmarks, ASIA and ALARM, were used to carry out the comparison. Results show that score metrics with hyperparameters that strength the tendency to select simpler network structures are better than score metrics with weaker tendency to select simpler network structures for both metrics (Heckerman-Geiger and modified MDL). Heckerman-Geiger Bayesian score metric works better than MDL with large datasets and MDL works better than Heckerman-Geiger with small datasets. The modified MDL gives similar results to Heckerman-Geiger for large datasets and close results to MDL for small datasets with stronger tendency to select simpler network structures

Propostas de solução para o problema de corte de estoque bidimensional de uma fábrica de móveis de pequeno porte

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Approximation Algorithms for Survivable Multicommodity Flow Problems with Applications to Network Design

Multicommodity flow (MF) problems have a wide variety of applications in areas such as VLSI circuit design, network design, etc., and are therefore very well studied. The fractional MF problems are polynomial time solvable while integer versions are NP-complete. However, exact algorithms to solve the fractional MF problems have high computational complexity. Therefore approximation algorithms to solve the fractional MF problems have been explored in the literature to reduce their computational complexity. Using these approximation algorithms and the randomized rounding technique, polynomial time approximation algorithms have been explored in the literature. In the design of high-speed networks, such as optical wavelength division multiplexing (WDM) networks, providing survivability carries great significance. Survivability is the ability of the network to recover from failures. It further increases the complexity of network design and presents network designers with more formidable challenges. In this work we formulate the survivable versions of the MF problems. We build approximation algorithms for the survivable multicommodity flow (SMF) problems based on the framework of the approximation algorithms for the MF problems presented in [1] and [2]. We discuss applications of the SMF problems to solve survivable routing in capacitated networks.

Maximum Series-Parallel Subgraph

Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1/2-approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n-1 edges and any series-parallel graph on n vertices has at most 2n-3 edges. We present a 7/12 -approximation for this problem and results showing the limits of our approach.

Mapping virtual networks onto substrate networks

Network virtualization is a promising technique for building the Internet of the future since it enables the low cost introduction of new features into network elements. An open issue in such virtualization is how to effect an efficient mapping of virtual network elements onto those of the existing physical network, also called the substrate network. Mapping is an NP-hard problem and existing solutions ignore various real network characteristics in order to solve the problem in a reasonable time frame. This paper introduces new algorithms to solve this problem based on 0–1 integer linear programming, algorithms based on a whole new set of network parameters not taken into account by previous proposals. Approximative algorithms proposed here allow the mapping of virtual networks on large network substrates. Simulation experiments give evidence of the efficiency of the proposed algorithms.

Lower complexity bounds in justification logic

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in~NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics are $\Pi^p_2$-hard, reproving and generalizing an earlier result by Milnikel.

A study of the time complexity of an Elitist Genetic Algorithm used for associating optical observations of MEO and GEO Space Debris

Currently several thousands of objects are being tracked in the MEO and GEO regions through optical means. The problem faced in this framework is that of Multiple Target Tracking (MTT). In this context both the correct associations among the observations, and the orbits of the objects have to be determined. The complexity of the MTT problem is defined by its dimension S. Where S stands for the number of ’fences’ used in the problem, each fence consists of a set of observations that all originate from dierent targets. For a dimension of S ˃ the MTT problem becomes NP-hard. As of now no algorithm exists that can solve an NP-hard problem in an optimal manner within a reasonable (polynomial) computation time. However, there are algorithms that can approximate the solution with a realistic computational e ort. To this end an Elitist Genetic Algorithm is implemented to approximately solve the S ˃ MTT problem in an e cient manner. Its complexity is studied and it is found that an approximate solution can be obtained in a polynomial time. With the advent of improved sensors and a heightened interest in the problem of space debris, it is expected that the number of tracked objects will grow by an order of magnitude in the near future. This research aims to provide a method that can treat the correlation and orbit determination problems simultaneously, and is able to e ciently process large data sets with minimal manual intervention.

Temperature Sensor Placement Including Routing Overhead and Sampling Inaccuracies

Dynamic thermal management techniques require a collection of on-chip thermal sensors that imply a significant area and power overhead. Finding the optimum number of temperature monitors and their location on the chip surface to optimize accuracy is an NP-hard problem. In this work we improve the modeling of the problem by including area, power and networking constraints along with the consideration of three inaccuracy terms: spatial errors, sampling rate errors and monitor-inherent errors. The problem is solved by the simulated annealing algorithm. We apply the algorithm to a test case employing three different types of monitors to highlight the importance of the different metrics. Finally we present a case study of the Alpha 21364 processor under two different constraint scenarios.

An Architecture forRepresenting Biological Processes based on Networks of Bio-inspired Processors

n this paper we propose the use of Networks of Bio-inspired Processors (NBP) to model some biological phenomena within a computational framework. In particular, we propose the use of an extension of NBP named Network Evolutionary Processors Transducers to simulate chemical transformations of substances. Within a biological process, chemical transformations of substances are basic operations in the change of the state of the cell. Previously, it has been proved that NBP are computationally complete, that is, they are able to solve NP complete problems in linear time, using massively parallel computations. In addition, we propose a multilayer architecture that will allow us to design models of biological processes related to cellular communication as well as their implications in the metabolic pathways. Subsequently, these models can be applied not only to biological-cellular instances but, possibly, also to configure instances of interactive processes in many other fields like population interactions, ecological trophic networks, in dustrial ecosystems, etc.