167 research outputs found
Opening the system to the environment: new theories and tools in classical and quantum settings
The thesis is organized as follows. Section 2 is a first, unconventional, approach to the topic of EPs. Having grown interest in the topic of combinatorics and graph theory, I wanted to exploit its very abstract and mathematical tools to reinterpret something very physical, that is, the EPs in wave scattering. To do this, I build the interpretation of scattering events from a graph theory perspective and show how EPs can be understood within this interpretation. In Section 3, I move from a completely classical treatment to a purely quantum one. In this section, I consider two quantum resonators coupled to two baths and study their dynamics with local and global master equations. Here, the EPs are the key physical features used as a witness of validity of the master equation. Choosing the wrong master equation in the regime of interest can indeed mask physical and fundamental features of the system. In Section 4, there are no EPs. However I transition towards a classical/quantum framework via the topic of open systems. My main contribution in this work is the classical stochastic treatment and simulation of a spin coupled to a bath. In this work, I show how a natural quantum--to--classical transition occurs at all coupling strengths when certain limits of spin length are taken. As a key result, I also show how the coupling to the environment in this stochastic framework induces a classical counterpart to quantum coherences in equilibrium. After this last topic, in Section 5, I briefly present the key features of the code I built (and later extended) for the latter project. This, in the form of a Julia registry package named SpiDy.jl, has seen further applications in branching projects and allows for further exploration of the theoretical framework. Finally, I conclude with a discussion section (see Sec. 5) where I recap the different conclusions gathered in the previous sections and propose several possible directions.Engineering and Physical Sciences Research Council (EPSRC
Planar Disjoint Paths, Treewidth, and Kernels
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . We
study the problem through the lens of kernelization, aiming at efficiently
reducing the input size in terms of a parameter. We show that Planar Disjoint
Paths does not admit a polynomial kernel when parameterized by unless coNP
NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e},
Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel
unless the WK-hierarchy collapses. Our reduction carries over to the setting of
edge-disjoint paths, where the kernelization status remained open even in
general graphs.
On the positive side, we present a polynomial kernel for Planar Disjoint
Paths parameterized by , where denotes the treewidth of the input
graph. As a consequence of both our results, we rule out the possibility of a
polynomial-time (Turing) treewidth reduction to under the same
assumptions. To the best of our knowledge, this is the first hardness result of
this kind. Finally, combining our kernel with the known techniques [Adler,
Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver,
SICOMP'94] yields an alternative (and arguably simpler) proof that Planar
Disjoint Paths can be solved in time , matching the
result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure
A Survey on Causal Discovery: Theory and Practice
Understanding the laws that govern a phenomenon is the core of scientific
progress. This is especially true when the goal is to model the interplay
between different aspects in a causal fashion. Indeed, causal inference itself
is specifically designed to quantify the underlying relationships that connect
a cause to its effect. Causal discovery is a branch of the broader field of
causality in which causal graphs is recovered from data (whenever possible),
enabling the identification and estimation of causal effects. In this paper, we
explore recent advancements in a unified manner, provide a consistent overview
of existing algorithms developed under different settings, report useful tools
and data, present real-world applications to understand why and how these
methods can be fruitfully exploited
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs
We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with O?(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in O?(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in O?(n?m) worst-case time and queries in O(?m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008]
Unweaving complex reactivity: graph-based tools to handle chemical reaction networks
La informació a nivell molecular obtinguda mitjançant estudis "in silico" s’ha establert com una eina essencial per a la
caracterització de mecanismes de reacció complexos. A més, l’aplicabilitat de la química computacional s’ha vist
substancialment ampliada a causa de l’increment continuat de la potència de càlcul disponible durant les darreres
dècades. Així, no només han augmentat la precisió dels mètodes a utilitzar o la mida dels sistemes a modelitzar sinó
també el grau de detall que es pot aconseguir en les descripcions mecanístiques resultants. Tanmateix, aquestes
caracteritzacions més profundes, usualment assistides per tècniques d’automatització que permeten l’exploració de
regions més extenses de l’espai químic, suposen un increment de la complexitat dels sistemes estudiats i per tant una
limitació de la seva interpretabilitat. En aquesta Tesi s’han proposat, desenvolupat i posat a prova diverses eines amb
el fi de fer el processament d’aquest tipus de xarxes de reacció químiques (CRNs) més simple i millorar la comprensió
de processos reactius i catalítics complexos. Aquesta col·lecció d’eines té com fonament la utilització de grafs per
modelitzar les xarxes (CRNs) corresponents, per poder fer servir els mètodes de la Teoria de Grafs (cerca de camins,
isomorfismes...) en un context químic. Més concretament, aquestes eines inclouen amk-tools, una llibreria per a la
visualització interactiva de xarxes de reacció descobertes de manera automàtica, gTOFfee, per a l’aplicació del
"energy span model" pel càlcul de la freqüència de recanvi de cicles catalítics complexos calculats
computacionalment, i OntoRXN, una ontologia per descriure CRNs de forma semàntica, integrant la topologia de la
xarxa i la informació calculada en una única entitat organitzada segons els principis del "Semantic Data".La información a nivel molecular obtenida por medio de estudios "in silico" se ha convertido en una herramienta
indispensable para la caracterización y comprensión de mecanismos de reacción complejos. Asimismo, la
aplicabilidad de la química computacional se ha ampliado sustancialmente como consecuencia del continuo
incremento de la potencia de cálculo durante las últimas décadas. Así, no sólo han aumentado la precisión de los
métodos o el tamaño de los sistemas modelizables, sino también el grado de detalle en la descripción mecanística.
Sin embargo, aumentar la profundidad de la caracterización de un sistema químico, usualmente a través de técnicas
de automatización que permiten explorar ecciones más extensas del espacio químico, supone un aumento en la
complejidad de los sistemas resultantes, dificultando la interpretación de los resultados. En esta Tesis se han
propuesto, desarrollado y puesto a prueba distintas herramientas para simplificar el procesado de este tipo de redes
de reacción químicas (CRNs), con el fin de mejorar la comprensión de procesos reactivos y catalíticos complejos.
Este conjunto de herramientas se basa en el uso de grafos para modelizar las redes (CRNs) correspondientes, con tal
de poder emplear los métodos de la Teoría de Grafos (búsqueda de caminos, isomorfismos...) bajo un contexto
químico. Concretamente, estas herramientas incluyen amk-tools, para la visualización interactiva de redes de
reacción descubiertas automáticamente, gTOFfee, para la aplicación del “energy span model” para calcular la
frecuencia de recambio de ciclos catalíticos complejos caracterizados computacionalmente, y OntoRXN, una
ontología para describir CRNs de manera semántica, integrando la topología de la red y la información calculada en
una única entidad organizada bajo los principios del “Semantic Data”.The molecular-level insights gathered through "in silico" studies have become an essential asset for the elucidation
and understanding of complex reaction mechanisms. Indeed, the applicability of computational chemistry has strongly
widened due to the vast increase in computational power along the last decades. In this sense, not only the accuracy
of the applied methods or the size of the target systems have increased, but also the level of detail attained for the
mechanistic description. However, performing deeper descriptions of chemical systems, most often resorting to
automation techniques that allow to easily explore larger parts of the chemical space, comes at the cost of also
augmenting their complexity, rendering the results much harder to interpret. Throughout this Thesis, we have
proposed, developed and tested a collection of tools aiming to process this kind of complex chemical reaction
networks (CRNs), in order to provide new insights on reactive and catalytic processes. All of these tools employ
graphs to model the target CRNs, in order to be able to use the methods of Graph Theory (e.g. path searches,
isomorphisms...) in a chemical context. The tools that are discussed include amk-tools, a framework for the interactive
visualization of automatically discovered reaction networks, gTOFfee, for the application of the energy span model to
compute the turnover frequency of computationally characterized catalytic cycles, and OntoRXN, an ontology for the
description of CRNs in a semantic manner integrating network topology and calculation information in a single,
highly-structured entity
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
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