96 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On Vizing's problem for triangle-free graphs

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    We prove that χ(G)≀⌈(Δ+1)/2⌉+1\chi(G) \le \lceil (\Delta+1)/2\rceil+1 for any triangle-free graph GG of maximum degree Δ\Delta provided Δ≄524\Delta \ge 524. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.Comment: 10 page

    Parameterized Graph Modification Beyond the Natural Parameter

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Asymptotics for Palette Sparsification

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    It is shown that the following holds for each Δ>0\varepsilon>0. For GG an nn-vertex graph of maximum degree DD and "lists" LvL_v (v∈V(G)v \in V(G)) chosen independently and uniformly from the ((1+Δ)ln⁥n(1+\varepsilon)\ln n)-subsets of {1,...,D+1}\{1, ..., D+1\}, G admits a proper coloring σ with σv∈Lv∀v G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v with probability tending to 1 as D→∞D \to \infty. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.Comment: 29 page

    Parameterized Graph Modification Beyond the Natural Parameter

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    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Investigating generalized Kitaev magnets using machine learning

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    Frustration in Kitaev-Materialien fĂŒhrt zu einem sehr reichhaltigen und komplexen Phasendiagramm, einschließlich der klassischen SpinflĂŒssigkeitsphase. Die Suche nach und das VerstĂ€ndnis von SpinflĂŒssigkeiten und weiteren neuartigen komplexen Phasen der Materie stehen im Mittelpunkt der heutigen Forschung zu kondensierter Materie. Mittels analytischer Methoden Ordnungsparameter zur Charakterisierung dieser Phasen zu finden, ist nahezu unmöglich. Bei niedrigen Temperaturen ordnen sich die meisten klassischen Spinsysteme in komplizierte Strukturen, die große magnetische Elementarzellen belegen, was die KomplexitĂ€t des Problems noch weiter erhöht und außerhalb des Anwendungsbereichs der meisten herkömmlichen Methoden liegt. In dieser Arbeit untersuchen wir die Hamilton-Operatoren realitĂ€tsnaher Kitaev-Materialien mithilfe maschinellen Lernens. Hauptmerkmale des zugrundeliegenden Algorithmus sind unbeaufsichtigtes Lernen, welches ermöglicht die Topologie eines Phasendiagramms ohne jegliche Vorkenntnisse erforschen, und Interpretierbarkeit, welche zur Analyse der Struktur der klassischen GrundzustĂ€nde notwendig ist. In den ersten drei Kapiteln werden wir den Algorithmus des maschinellen Lernens auf verschiedene Hamilton-Operatoren anwenden, die zur Modellierung von Kitaev-Materialien eingesetzt werden, um zu untersuchen inwieweit die Quantenmodelle und die experimentellen Beobachtungen allein durch deren klassischen Grenzfall erklĂ€rt werden können. DarĂŒber hinaus erforschen wir weitere Features dieses Algorithmus, die es uns ermöglichen, verborgene Symmetrien, lokale EinschrĂ€nkungen der klassischen SpinflĂŒssigkeiten, sowie bisher unbekannte Phasen im hochdimensionalen Phasenraum aufzudecken. In den letzten beiden Kapiteln werden wir uns mit dem VerstĂ€ndnis der Struktur der klassischen GrundzustĂ€nde befassen, welche durch die Verflechtung mehrerer Helices charakterisiert sind. Wir werden auch versuchen, die Signatur dieser Phasen in Experimenten zu verstehen, indem wir die Dynamik und den Transport durch Kitaev-Magnete untersuchen. Diese Arbeit beweist die Tauglichkeit von maschinellem Lernen, hochkomplexe Phasendiagramme mit wenig bis gar keinem Vorwissen aufzudecken und hochfrustrierten Magnetismus zu erforschen. Die Kombination aus maschinellem und menschlichem Einsatz ebnet den Weg zu neuen und spannenden physikalischen Erkenntnissen.Bond frustration in Kitaev materials leads to a very rich phase diagram with highly intricate phases including the classical spin liquid phase. The search and understanding of spin liquids and novel complex phases of matter is at the heart pf present day condensed matter research. To search and design order parameters to characterize these phases using analytical approaches is a nearly impossible task. At low temperatures, most of the classical spins order into complicated spin structures occupying large magnetic unit cells which further adds to the complication and is out of the realm of most traditional methods. In this thesis we investigate realistic Kitaev material Hamiltonians using a machine learning framework whose key features, of unsupervised learning which helps us study the topology of the phase diagram without prior knowledge and interpretability which helps us analyse the structure of the classical ground states, are exploited. In the first three chapters, we shall use this framework on different Hamiltonians used to model Kitaev materials and understand to what extent the quantum limit and experimental results could be explained just by the classical limit of these models. We in addition explore other features of this framework which lets us uncover hidden symmetries as well as local constraints for the classical spin liquids and hitherto unreported new phases in the high dimensional phase space. In the last two chapters we shall dwell on the understanding the structure of the classical ground states which is quite complicated as it hosts a tangle of multiple helices. We shall also try and understand the signature of these phases on experiments by studying the dynamics and transport through Kitaev magnets thus bridging the gap between experiment and theory. This thesis proves instances of using machine learning to uncover highly complex phase diagrams with little to no previous knowledge and serve as a paradigm to explore highly frustrated magnetism. Through a combination of machine and human effort we are on the way to uncover new and exciting physics

    Uniformly Random Colourings of Sparse Graphs

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    We analyse uniformly random proper kk-colourings of sparse graphs with maximum degree Δ\Delta in the regime Δ<kln⁡k\Delta < k\ln k . This regime corresponds to the lower side of the shattering threshold for random graph colouring, a paradigmatic example of the shattering threshold for random Constraint Satisfaction Problems. We prove a variety of results about the solution space geometry of colourings of fixed graphs, generalising work of Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the performance of stochastic local search algorithms in this regime. Our central proof relies only on elementary techniques, namely the first-moment method and a quantitative induction, yet it strengthens list-colouring results due to Vu, and more recently Davies, Kang, P., and Sereni, and generalises state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It further yields an approximately tight lower bound on the number of colourings, also known as the partition function of the Potts model, with implications for efficient approximate counting
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