On the Complexity of the Median and Closest Permutation Problems

Genome rearrangements are events where large blocks of DNA exchange places during evolution. The analysis of these events is a promising tool for understanding evolutionary genomics, providing data for phylogenetic reconstruction based on genome rearrangement measures. Many pairwise rearrangement distances have been proposed, based on finding the minimum number of rearrangement events to transform one genome into the other, using some predefined operation. When more than two genomes are considered, we have the more challenging problem of rearrangement-based phylogeny reconstruction. Given a set of genomes and a distance notion, there are at least two natural ways to define the "target" genome. On the one hand, finding a genome that minimizes the sum of the distances from this to any other, called the median genome. Finding a genome that minimizes the maximum distance to any other, called the closest genome. Considering genomes as permutations, some distance metrics have been extensively studied. We investigate median and closest problems on permutations over the metrics: breakpoint, swap, block-interchange, short-block-move, and transposition. In biological matters some values are usually small, such as the solution value d or the number k of input permutations. For each of these metrics and parameters d or k, we analyze the closest and the median problems from the viewpoint of parameterized complexity. We obtain the following results: NP-hardness for finding the median/closest permutation for some metrics, even for k = 3; Polynomial kernels for the problems of finding the median permutation of all studied metrics, considering the target distance d as parameter; NP-hardness result for finding the closest permutation by short-block-moves; FPT algorithms and infeasibility of polynomial kernels for finding the closest permutation for some metrics parameterized by the target distance d

Rearranjo de genomas : algoritmos e complexidade

This thesis discusses events of genome rearrangements problems: transposition, breakpoint, block interchange, short block move, and the restricted multi break. We consider problems of sorting, closest permutation, and the diameter. We develop approximation algorithms, NP-completeness and properties about these problems. Regarding the sorting by transpositions, which is an NP-complete problem, several approximation algorithms were proposed based on the graph called the reality and desire diagram. Through a case analyses of the cycles of this graph, we propose a new one which achieves so far the best 1.375 ratio and O(n log n) running time complexity. Although sorting by transpositions is NP-complete, there are several metrics whose sorting problems are polynomial or are open. In such cases, an interesting problem arises to find a permutation with maximum distance of an input permutation set at most some value, this is the closest permutation problem. We show that with respect to the polynomial distance problems of breakpoint and of block interchange, both problems are NP-complete. In order to explore properties on operations that are restriction or generalization of others, we deal with the operation of short block move and we propose the operation of restricted multi break. Regarding the short block move, we show tractable classes of permutations, properties on the permutation graph, and we show that the closest permutation problem is NP-complete. Regarding the restricted multi break, we study two versions: one where the number of non reversible blocks is bounded by a constant, and another one whose number of non reversible blocks is arbitrary. We prove tight bounds on the distance and the diameter problems for both versions.Esta tese trata de rearranjo de genomas nos eventos de: transposição, pontos de quebra, movimento de blocos, movimento de blocos curtos, e de multi corte restritos. Abordamos os problemas de ordenação, permutação mais próxima, e de diâmetro. Apresentamos algoritmos aproximativos, NP-completudes e propriedades. Sobre o problema de ordenação por transposições, provado ser NP-completo, alguns algoritmos aproximativos foram propostos baseados no grafo chamado diagrama de realidade e desejo. Através da análise dos ciclos deste grafo, propomos um novo algoritmo que atinge melhores resultados correntes, tanto de razão de aproximação de 1,375 quanto de complexidade de tempo de O(n log n). Embora ordenação por transposições seja NP-completo, há outros problemas polinomiais ou em aberto. Nestes casos, surge o desafio de encontrar uma permutação que esteja a uma distância máxima limitada por algum valor em relação a um conjunto de permutações dadas de entrada. Este é o problema de encontrar a permutação mais próxima. Mostramos que, em relação `as operações de pontos de quebra e de movimento de blocos, tais problemas são NP-completos. Com o objetivo de obter propriedades sobre operações que restingem ou generalizam outras, tratamos da operação de movimento de blocos curtos e propomos a operação de multi corte restritos. Sobre movimento de blocos curtos, mostramos classes com distâncias exatas, propriedades sobre o grafo de permutação, e mostramos que o problema de permutação mais próxima é NP-completo. Sobre multi corte restritos, tratamos de duas variações: uma cujo número de blocos não reversíveis é limitado por constante, e outra cujo número de blocos não reversíveis é arbitrário. Mostramos limites justos de distância e de diâmetro para ambas as versões

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Encoding correlations in open quantum systems

The study of open quantum systems (OQS) is a vast topic consisting of diverse approaches being, or having been, carried out in divers places and times. One of the themes of this work is to demonstrate the equivalence of a number of apparently different techniques in order to travel a small way along the path of unifying the study of OQS. We present the fixed mode method (FMM) for deriving mappings between linear Hamiltonians and show how it can be used to verify the pre-eminence of the spectral density in the study of open quantum systems. We utilise the FMM to derive the collective mode method (CMM) and show that it encompasses the reaction coordinate (RC) mapping, the chain mapping, Fano diagonalisation (FD), and (when combined with a bath combination (BC) mapping) some varieties of the technique of discretisation within its scope. We demonstrate the versatility of the CMM by applying it to the study of ground states of the spin boson model (SBM) and dynamics of fermionic systems. Utilising a discretised multipolaron ansatz (MP); and exact diagonalisation and master equation techniques respectively. Another theme is that of the computation. By their very nature OQS require large scale computations, owing to the need to characterise correlations between the systems of interest (S) and the (effectively infinite) wider world. This means that many problems are now beyond the scope of humanity's (inbuilt) computational capacity and rely on the use of digital processors. As such it becomes increasingly important to utilise the intuition of humans and the speed of machines in an efficient manner. Without due care we could end up either producing unmanagable quantities of poorly understood data; or spend an eternity dealing with the minutiae of a derivation. With this in mind we attempt to present our results in a form that they might be efficiently encoded. That is, along with the methods themselves, we offer suggestions as to how such methods could be efficiently implemented and how they might be useful in pedagogical and academic environments. Throughout this work we have made use of continuum representations of particulate environments (or baths), moving to discretised versions only when necessitated by the constraints of a particular problem or methodology. We hope to demonstrate how thinking in terms of the continuum can aid intuition and calculations. Finally, wherever possible, we have attempted to provide `proof of principal' for our assertions. We have applied the techniques we derive and investigate (FMM, CMM, MP) to particular models in order to provide a detailed (but far from exhaustive) demonstration of how such methods could be used. The aim being that intuition developed from the `toy models' we investigate can be applied more widely to experimentally motivated problems within the field.Open Acces

Engineering Education and Research Using MATLAB

MATLAB is a software package used primarily in the field of engineering for signal processing, numerical data analysis, modeling, programming, simulation, and computer graphic visualization. In the last few years, it has become widely accepted as an efficient tool, and, therefore, its use has significantly increased in scientific communities and academic institutions. This book consists of 20 chapters presenting research works using MATLAB tools. Chapters include techniques for programming and developing Graphical User Interfaces (GUIs), dynamic systems, electric machines, signal and image processing, power electronics, mixed signal circuits, genetic programming, digital watermarking, control systems, time-series regression modeling, and artificial neural networks

Colliders And Neutrinos

This book is a collection of theoretical advanced summer institute lectures by world experts in the field of collider physics and neutrinos, the two frontier areas of particle physics today. It is aimed at graduate students and beginning researchers, and as such, provides many pedagogical details not generally available in standard conference proceedings