Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

Reliable postprocessing improvement of van der Waals heterostructures

The successful assembly of heterostructures consisting of several layers of different 2D materials in arbitrary order by exploiting van der Waals forces has truly been a game changer in the field of low dimensional physics. For instance, the encapsulation of graphene or MoS2 between atomically flat hexagonal boron nitride (hBN) layers with strong affinity and graphitic gates that screen charge impurity disorder provided access to a plethora of interesting physical phenomena by drastically boosting the device quality. The encapsulation is accompanied by a self-cleansing effect at the interfaces. The otherwise predominant charged impurity disorder is minimized and random strain fluctuations ultimately constitute the main source of residual disorder. Despite these advances, the fabricated heterostructures still vary notably in their performance. While some achieve record mobilities, others only possess mediocre quality. Here, we report a reliable method to improve fully completed van der Waals heterostructure devices with a straightforward post-processing surface treatment based on thermal annealing and contact mode AFM. The impact is demonstrated by comparing magnetotransport measurements before and after the AFM treatment on one and the same device as well as on a larger set of treated and untreated devices to collect device statistics. Both the low temperature properties as well as the room temperature electrical characteristics, as relevant for applications, improve on average substantially. We surmise that the main beneficial effect arises from reducing nanometer scale corrugations at the interfaces, i.e. the detrimental impact of random strain fluctuations

On the Classification of Legendrian Rational Tangles

We show that under certain conditions the flyping operation on rational tangles, which produces topologically isotopic tangles, may also produce tangles which are not Legendrian isotopic when viewed in the standard contact structure on $\mathbb{R}^3$. This work is motivated by questions posed by Traynor, and incorporates techniques inspired by work of Eliashberg and Fraser. The proofs within employ a new method of diagramming rational tangles which is also used to easily describe the characteristic foliations of compressing discs for their complements.Comment: 44 pages, 37 figure

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On Using Graphical Calculi: Centers, Zeroth Hochschild Homology and Possible Compositions of Induction and Restriction Functors in Various Diagrammatical Algebras

This thesis is divided into three chapters, each using certain graphical calculus in a slightly different way. In the first chapter, we compute the dimension of the center of the 0-Hecke algebra Hn and of the Nilcoxeter algebra NCn using a calculus of diagrams on the Moebius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for Z(NCn) and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for Z(Hn) can also be obtained in a specic way from this topological calculus. In the second chapter, we also use a calculus of diagrams on the annulus and the Moebius band to determine the zeroth Hochschild Homology of Kuperberg's webs for rank two Lie algebras. We use results from Sikora and Westbury to prove the linear independence of these webs on these surfaces. In the third chapter, we use other diagrams to attempt to find explicitely the possible compositions of the induction and restriction functors in the cyclotomic quotients of the NilHecke algebra. We use a computer program to obtain partial results