615 research outputs found

    Opening the system to the environment: new theories and tools in classical and quantum settings

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    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

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A proof of the Ryser-Brualdi-Stein conjecture for large even nn

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    A Latin square of order nn is an nn by nn grid filled using nn symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order nn contains a transversal with n−1n-1 cells, and a transversal with nn cells if nn is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order nn has a transversal with n−O(log⁡n/log⁡log⁡n)n-O(\log n/\log\log n) cells. Here, we show, for sufficiently large nn, that every Latin square of order nn has a transversal with n−1n-1 cells. We also apply our methods to show that, for sufficiently large nn, every Steiner triple system of order nn has a matching containing at least (n−4)/3(n-4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3−O(log⁡n/log⁡log⁡n)n/3-O(\log n/\log\log n) edges, and proves a conjecture of Brouwer from 1981 for large nn.Comment: 71 pages, 13 figure

    A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

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    Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

    Complexity Framework for Forbidden Subgraphs II: When Hardness Is Not Preserved under Edge Subdivision

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    For a fixed set H{\cal H} of graphs, a graph GG is H{\cal H}-subgraph-free if GG does not contain any H∈HH \in {\cal H} as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on H{\cal H}-subgraph-free graphs (for finite sets H{\cal H}) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity H{\cal H}-subgraph-free graphs is unknown. In this paper, we study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: kk-Induced Disjoint Paths, C5C_5-Colouring, Hamilton Cycle and Star 33-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and differs from problems that do satisfy all three conditions of the framework. Hence, we exhibit a rich complexity landscape among problems for H{\cal H}-subgraph-free graph classes

    Flat bands of periodic graphs

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    We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands, some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell and we compute the list of such bands for small cells. We next prove that flat bands are rare and vanish under arbitrarily small perturbations by periodic potentials. Additional folklore results are proved and many questions are still open.Comment: 25 pages, 19 figure

    Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

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    Two graphs GG and HH are homomorphism indistinguishable over a class of graphs F\mathcal{F} if for all graphs F∈FF \in \mathcal{F} the number of homomorphisms from FF to GG is equal to the number of homomorphisms from FF to HH. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl

    Efficient parameterized algorithms on structured graphs

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    In der klassischen KomplexitĂ€tstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhĂ€ngig von der EingabegrĂ¶ĂŸe angegeben. In dem Kontext der parametrisierten KomplexitĂ€tstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusĂ€tzlich zu der EingabengrĂ¶ĂŸe noch einen Parameter berĂŒcksichtigt, welcher angibt, wie strukturiert die Eingabe bezĂŒglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist. Der erste Hauptteil dieser Arbeit fĂŒhrt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm fĂŒr den grĂ¶ĂŸtmöglichen Parameterwert ĂŒbereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und ĂŒbertreffen diese bereits fĂŒr leicht nichttriviale Parameterwerte. Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen AusdrĂŒcken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen ĂŒbereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value. In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems. Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values. As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure. In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures. Using algebraic expressions, we define new combined graph classes of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout

    Generating Polynomials of Exponential Random Graphs

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    The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM). In the past, determining when a probability distribution has strong negative dependence has proven to be difficult ([Pem00, BBL09]). The negative dependence of a probability distribution is characterized by properties of its corresponding generating polynomial ([BBL09]). This thesis bridges the theory of exponential random graphs with the geometry of their generating polynomials, namely, when and how they satisfy the stable or Lorentzian properties ([Wag09, BBL09, BH20, AGV21]). We provide necessary and sufficient conditions as well as full characterizations of the parameter space for when this model has a stable or Lorentzian generating polynomial. This is done using a well-developed dictionary between probability distributions and their corresponding multiaffine generating polynomials. In particular, we characterize when the generating polynomial of a random graph model with a large symmetry group is irreducible. We assert that the edge parameter of the exponential random graph model does not affect stability and that the triangle and k-star parameters are necessarily related if the model is stable or Lorentzian. We also provide full Lorentzian and stable characterizations for the model on K3 and a Lorentzian characterization for specializations of the model on K4

    Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

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    Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in its (open) neighborhood. Typical examples of graphs with large proper conflict-free chromatic number include graphs with large chromatic number and bipartite graphs isomorphic to the 11-subdivision of graphs with large chromatic number. In this paper, we prove two rough converse statements that hold even in the list-coloring setting. The first is for sparse graphs: for every graph HH, there exists an integer cHc_H such that every graph with no subdivision of HH is (properly) conflict-free cHc_H-choosable. The second applies to dense graphs: every graph with large conflict-free choice number either contains a large complete graph as an odd minor or contains a bipartite induced subgraph that has large conflict-free choice number. These give two incomparable (partial) answers of a question of Caro, Petru\v{s}evski and \v{S}krekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about proper conflict-free coloring and odd coloring in the literature. Moreover, we prove that every graph with layered treewidth at most ww is (properly) conflict-free (8w−1)(8w-1)-choosable. This result applies to (g,k)(g,k)-planar graphs, which are graphs whose coloring problems have attracted attention recently.Comment: Hickingbotham recently independently announced a paper (arXiv:2203.10402) proving a result similar to the ones in this paper. Please see the notes at the end of this paper for details. v2: add results for odd minors, which applies to graphs with unbounded degeneracy, and change the title of the pape
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