On the Crossing Number of the Cartesian Product of a Sunlet Graph and a Star Graph

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given

Upward planarization and layout

Die Visualisierung von gerichteten azyklischen Graphen (DAGs) gehört zu den wichtigsten Aufgaben im automatischen Zeichnen von Graphen. Hierbei suchen wir für einen gegebenen DAG G eine Zeichnung von G (Aufwärtszeichnung von G genannt), sodass alle Kanten als Kurven streng monoton in vertikaler Richtung steigend gezeichnet werden. Um die Lesbarkeit der Zeichnung zu erhöhen, sollte neben der Aufwärtseigenschaft auch die Anzahl der Kantenkreuzungen in der Zeichnung möglichst gering sein. In dieser Dissertation entwerfen wir einen neuen Ansatz zur Visualisierung von gerichteten Graphen, der auf der Idee der Aufwärtsplanarisierung basiert. Wir stellen zuerst ein innovatives Aufwärtsplanarisierungverfahren vor, das neue Techniken für die Berechnung aufwärtsplanare Untergraphen und die anschließende Kanteneinfügephase einsetzt. Vor allem werden in dem neuen Verfahren keine Schichtungstechniken zur Kreuzungsminimierung benutzt, wie wir sie aus dem Zeichenverfahren von Sugiyama et al. [STT81] oder aus dem Aufwärtsplanarisierungsverfahren von Eiglsperger et al. [EKE03] kennen. Die Festlegung einer Schichtung kann nämlich zu sehr schlechten Ergebnissen führen. Folglich besitzt das neue Verfahren nicht die Nachteile der bisherigen Kreuzungsminimierungsverfahren. Experimentellen Analysen zeigen, dass das neue Aufwärtsplanarisierungsverfahren deutlich bessere Ergebnisse liefert als das klassische, auf Schichtungen basierende Kreuzungsminimierungsverfahren, und dies unabhängig von den benutzten Lösungsansätzen (heuristisch oder optimal) für die klevel Kreuzungsminimierungsphase. Auch im Vergleich mit den bekannten Aufwärtsplanarisierungsverfahren (Di Battista et al. [BPTT89] und Eiglsperger et al. [EKE03]) zeigt sich, dass der neue Ansatz weitaus bessere Ergebnisse liefert. Wir stellen auch zwei Erweiterungen des neuen Ansatzes vor: eine Erweiterung zur Aufwärtsplanarisierung von gerichteten Hypergraphen und eine zur Unterstützung von Port Constraints. Das Ergebnis der Aufwärtsplanarisierung ist eine aufwärtsplanare Repräsentation (UPR) — ein eingebetteter DAG, in dem Kreuzungen durch künstliche Dummy-Knoten modelliert werden. Wir stellen ein Layoutverfahren zur Realisierung solcher UPRs vor, d.h., ein Verfahren, das aus einem UPR eine Aufwärtszeichnung konstruiert, sodass die Kantenkreuzungen in der Zeichnung zu den Dummy-Knoten des gegebenen UPR korrespondieren. Die wenigen existierenden Zeichenverfahren zur Realisierung von UPRs sind sehr einfach und wurden ursprünglich entwickelt, um planare st-Graphen zu zeichnen. Unser neues Verfahren stellt somit das erste Layoutverfahren dar, das speziell im Hinblick auf die Realisierung von UPRs entworfen wurde. Es bietet zwei wichtige Vorteile gegenüber dem etablierten Standardzeichenalgorithmus von Sugiyama et al.: Die Zeichnungen besitzen wesentlich weniger Kreuzungen, was zur deutlichen Verbesserung der Lesbarkeit führt. Ferner sind sie strukturierter und machen einen aufgeräumteren Eindruck

Multi-objective tools for the vehicle routing problem with time windows

Most real-life problems involve the simultaneous optimisation of two or more, usually conflicting, objectives. Researchers have put a continuous effort into solving these problems in many different areas, such as engineering, finance and computer science. Over time, thanks to the increase in processing power, researchers have created methods which have become increasingly sophisticated. Most of these methods have been based on the notion of Pareto dominance, which assumes, sometimes erroneously, that the objectives have no known ranking of importance. The Vehicle Routing Problem with Time Windows (VRPTW) is a logistics problem which in real-life applications appears to be multi-objective. This problem consists of designing the optimal set of routes to serve a number of customers within certain time slots. Despite this problem’s high applicability to real-life domains (e.g. waste collection, fast-food delivery), most research in this area has been conducted with hand-made datasets. These datasets sometimes have a number of unrealistic features (e.g. the assumption that one unit of travel time corresponds to one unit of travel distance) and are therefore not adequate for the assessment of optimisers. Furthermore, very few studies have focused on the multi-objective nature of the VRPTW. That is, very few have studied how the optimisation of one objective affects the others. This thesis proposes a number of novel tools (methods + dataset) to address the above- mentioned challenges: 1) an agent-based framework for cooperative search, 2) a novel multi-objective ranking approach, 3) a new dataset for the VRPTW, 4) a study of the pair-wise relationships between five common objectives in VRPTW, and 5) a simplified Multi-objective Discrete Particle Swarm Optimisation for the VRPTW

Multi-objective tools for the vehicle routing problem with time windows

Most real-life problems involve the simultaneous optimisation of two or more, usually conflicting, objectives. Researchers have put a continuous effort into solving these problems in many different areas, such as engineering, finance and computer science. Over time, thanks to the increase in processing power, researchers have created methods which have become increasingly sophisticated. Most of these methods have been based on the notion of Pareto dominance, which assumes, sometimes erroneously, that the objectives have no known ranking of importance. The Vehicle Routing Problem with Time Windows (VRPTW) is a logistics problem which in real-life applications appears to be multi-objective. This problem consists of designing the optimal set of routes to serve a number of customers within certain time slots. Despite this problem’s high applicability to real-life domains (e.g. waste collection, fast-food delivery), most research in this area has been conducted with hand-made datasets. These datasets sometimes have a number of unrealistic features (e.g. the assumption that one unit of travel time corresponds to one unit of travel distance) and are therefore not adequate for the assessment of optimisers. Furthermore, very few studies have focused on the multi-objective nature of the VRPTW. That is, very few have studied how the optimisation of one objective affects the others. This thesis proposes a number of novel tools (methods + dataset) to address the above- mentioned challenges: 1) an agent-based framework for cooperative search, 2) a novel multi-objective ranking approach, 3) a new dataset for the VRPTW, 4) a study of the pair-wise relationships between five common objectives in VRPTW, and 5) a simplified Multi-objective Discrete Particle Swarm Optimisation for the VRPTW

Planning for steerable needles in neurosurgery

The increasing adoption of robotic-assisted surgery has opened up the possibility to control innovative dexterous tools to improve patient outcomes in a minimally invasive way. Steerable needles belong to this category, and their potential has been recognised in various surgical fields, including neurosurgery. However, planning for steerable catheters' insertions might appear counterintuitive even for expert clinicians. Strategies and tools to aid the surgeon in selecting a feasible trajectory to follow and methods to assist them intra-operatively during the insertion process are currently of great interest as they could accelerate steerable needles' translation from research to practical use. However, existing computer-assisted planning (CAP) algorithms are often limited in their ability to meet both operational and kinematic constraints in the context of precise neurosurgery, due to its demanding surgical conditions and highly complex environment. The research contributions in this thesis relate to understanding the existing gap in planning curved insertions for steerable needles and implementing intelligent CAP techniques to use in the context of neurosurgery. Among this thesis contributions showcase (i) the development of a pre-operative CAP for precise neurosurgery applications able to generate optimised paths at a safe distance from brain sensitive structures while meeting steerable needles kinematic constraints; (ii) the development of an intra-operative CAP able to adjust the current insertion path with high stability while compensating for online tissue deformation; (iii) the integration of both methods into a commercial user front-end interface (NeuroInspire, Renishaw plc.) tested during a series of user-controlled needle steering animal trials, demonstrating successful targeting performances. (iv) investigating the use of steerable needles in the context of laser interstitial thermal therapy (LiTT) for maesial temporal lobe epilepsy patients and proposing the first LiTT CAP for steerable needles within this context. The thesis concludes with a discussion of these contributions and suggestions for future work.Open Acces

Algorithms for visualization of graph-based structures

Buildings today are built to maintain a healthy indoor environment and an efficient energy usage which is probably why damages caused by dampness has increased since the 1960’s. A study between year 2008 and 2010 showed that 26 percent of the 110 000 examined houses had damages and flaws caused by dampness that could prove to be harmful later on. This means that one out of four bathrooms risk the chance to develop damages by dampness. Approximately 2 percent of the houses had already developed water damages. It is here where the problems appear. A house or a building that is damaged by water of dampness need time to dry out before any renovation can take place. This means that damaged parts must be removed and allowed to dry out, this takes a long time to do and the costs are high and at the same time it can cause inconvenience to the residents. Here is where the Air Gap Method enters the picture. The meaning with the method is to drain and dry out the moisture without the need to perform a larger renovation. The Air Gap Method is a so called "forgiving"-system that is if water damages occur the consequences will be small. The Air Gap method means that an air gap is created in the walls, ceiling and the floor where a heating cable in the gap heats up the air and creates an air movement. The point is to create a stack effect in the gap that with the help of the air movement transports the damp air through an opening by the ceiling. The aim of this thesis is to examine if it’s necessary with the heating cable in the air gap and if there is a specific drying out pattern of the water damaged bathroom floor. The possibility of mould growth will also be examined. The study showed that the damped floor did dry out even without a heating cable, but as one of the studies showed signs of mould growth it is shown that the risk for mould growth is higher without a heating cable. There was a seven days difference in the drying out time between the studies with and without the heating cable; this difference can be decisive for mould growth which is why the heating cable is recommended. The Air Gap method is quite easy to apply in houses with light frame constructions simply by using a smaller dimension on the studs to create the air gap in the floor and walls. The method can also be applied in apartment buildings with a concrete frame by using the room-in- room principal. When renovating existing bathrooms it’s easier to use prefabricated elements to create the air gap in the floor and walls. ~