Change Actions: Models of Generalised Differentiation

Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the F\`aa di Bruno construction

Verification of Hierarchical Artifact Systems

Data-driven workflows, of which IBM's Business Artifacts are a prime exponent, have been successfully deployed in practice, adopted in industrial standards, and have spawned a rich body of research in academia, focused primarily on static analysis. The present work represents a significant advance on the problem of artifact verification, by considering a much richer and more realistic model than in previous work, incorporating core elements of IBM's successful Guard-Stage-Milestone model. In particular, the model features task hierarchy, concurrency, and richer artifact data. It also allows database key and foreign key dependencies, as well as arithmetic constraints. The results show decidability of verification and establish its complexity, making use of novel techniques including a hierarchy of Vector Addition Systems and a variant of quantifier elimination tailored to our context.Comment: Full version of the accepted PODS pape

A tale of two packing problems : improved algorithms and tighter bounds for online bin packing and the geometric knapsack problem

In this thesis, we deal with two packing problems: the online bin packing and the geometric knapsack problem. In online bin packing, the aim is to pack a given number of items of different size into a minimal number of containers. The items need to be packed one by one without knowing future items. For online bin packing in one dimension, we present a new family of algorithms that constitutes the first improvement over the previously best algorithm in almost 15 years. While the algorithmic ideas are intuitive, an elaborate analysis is required to prove its competitive ratio. We also give a lower bound for the competitive ratio of this family of algorithms. For online bin packing in higher dimensions, we discuss lower bounds for the competitive ratio and show that the ideas from the one-dimensional case cannot be easily transferred to obtain better two-dimensional algorithms. In the geometric knapsack problem, one aims to pack a maximum weight subset of given rectangles into one square container. For this problem, we consider online approximation algorithms. For geometric knapsack with square items, we improve the running time of the best known PTAS and obtain an EPTAS. This shows that large running times caused by some standard techniques for geometric packing problems are not always necessary and can be improved. Finally, we show how to use resource augmentation to compute optimal solutions in EPTAS-time, thereby improving upon the known PTAS for this case.In dieser Arbeit betrachten wir zwei Packungsprobleme: Online Bin Packing und das geometrische Rucksackproblem. Bei Online Bin Packing versucht man, eine gegebene Menge an Objekten verschiedener Größe in die kleinstmögliche Anzahl an Behältern zu packen. Die Objekte müssen eins nach dem anderen gepackt werden, ohne zukünftige Objekte zu kennen. Für eindimensionales Online Bin Packing beschreiben wir einen neuen Algorithmus, der die erste Verbesserung gegenüber dem bisher besten Algorithmus seit fast 15 Jahren darstellt. Während die algorithmischen Ideen intuitiv sind, ist eine ausgefeilte Analyse notwendig um das Kompetitivitätsverhältnis zu beweisen. Für Online Bin Packing in mehreren Dimensionen geben wir untere Schranken für das Kompetitivitätsverhältnis an und zeigen, dass die Ideen aus dem eindimensionalen Fall nicht direkt zu einer Verbesserung führen. Beim geometrischen Rucksackproblem ist es das Ziel, eine größtmögliche Teilmenge gegebener Rechtecke in einen einzelnen quadratischen Behälter zu packen. Für dieses Problem betrachten wir Approximationsalgorithmen. Für das Problem mit quadratischen Objekten verbessern wir die Laufzeit des bekannten PTAS zu einem EPTAS. Die langen Laufzeiten vieler Standardtechniken für geometrische Probleme können also vermieden werden. Schließlich zeigen wir, wie Ressourcenvergrößerung genutzt werden kann, um eine optimale Lösung in EPTAS-Zeit zu berechnen, was das bisherige PTAS verbessert.Google PhD Fellowshi