Half-Integer Winding Number Solutions to the Landau-Ginzburg-Higgs Equations and Instability of the Abrikosov-Nielsen-Olesen Vortex

New solutions to the abelian U(1) Higgs model, corresponding to vortices of integer and half-integer winding number bound onto the edges of domain walls and possibly surrounded by annular current flows, are described, based on a fine-grained analysis of the topology of such configurations in spacetime. The existence of these states, which saturate BPS bounds in specific limits and are quite reminiscent of D-branes and membranes in general, could have interesting and some important consequences in a wide range of physical contexts. For instance, they raise the possibility that for some regimes of couplings the usual vortex of unit winding number would split into two vortices each of one-half winding number bound by a domain wall. A similar approach may also be relevant to other known topological states of field theory.Comment: 52 pages (Latex) + 2 postscript figure

On the Stability of Distribution Topologies in Peer-to-Peer Live Streaming Systems

Peer-to-Peer Live-Streaming-Systeme sind ständigen Störungen ausgesetzt.Insbesondere ermöglichen unzuverlässige Teilnehmer Ausfälle und Angriffe, welche überraschend Peers aus dem System entfernen. Die Folgen solcher Vorfälle werden großteils von der Verteilungstopologie bestimmt, d.h. der Kommunikationsstruktur zwischen den Peers.In dieser Arbeit analysieren wir Optimierungsprobleme welche bei der Betrachtung von Stabilitätsbegriffen für solche Verteilungstopologien auftreten. Dabei werden sowohl Angriffe als auch unkoordinierte Ausfälle berücksichtigt.Zunächst untersuchen wir die Berechnungskomplexität und Approximierbarkeit des Problems resourcen-effiziente Angriffe zu bestimmen. Dies demonstriert Beschränkungen in den Planungsmöglichkeiten von Angreifern und zeigt inwieweit die Topologieparameter die Schwierigkeit solcher Angriffsrobleme beeinflussen. Anschließend studieren wir Topologieformationsprobleme. Dabei sind Topologieparameter vorgegeben und es muss eine passende Verteilungstopologie gefunden werden. Ziel ist es Topologien zu erzeugen, welche den durch Angriffe mit beliebigen Parametern erzeugbaren maximalen Schaden minimieren.Wir identifizieren notwendige und hinreichende Eigenschaften solcher Verteilungstopologien. Dies führt zu mathematisch fundierten Zielstellungen für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen.Wir zeigen zwei große Klassen effizient konstruierbarer Verteilungstopologien, welche den maximal möglichen, durch Angriffe verursachten Paketverlust minimieren. Zusätzlich beweisen wir, dass die Bestimmung dieser Eigenschaft für beliebige Topologien coNP-vollständig ist.Soll die maximale Anzahl von Peers minimiert werden, bei denen ein Angriff zu ungenügender Stream-Qualität führt, ändern sich die Anforderungen an Verteilungstopologien. Wir zeigen, dass dieses Topologieformationsproblem eng mit offenen Problemen aus Design- und Kodierungstheorie verwandt ist.Schließlich analysieren wir Verteilungstopologien die den durch unkoordinierte Ausfälle zu erwartetenden Paketverlust minimieren. Wir zeigen Eigenschaften und Existenzbedingungen. Außerdem bestimmen wir die Berechnungskomplexität des Auffindens solcher Topologien. Unsere Ergebnisse liefern Richtlinien für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen und zeigen auf, welche Stabilitätsziele effizient erreicht werden können.The stability of peer-to-peer live streaming systems is constantly challenged. Especially, the unreliability and vulnerability of their participants allows for failures and attacks suddenly disabling certain sets of peers. The consequences of such events are largely determined by the distribution topology, i.e., the pattern of communication between the peers.In this thesis, we analyze a broad range of optimization problems concerning the stability of distribution topologies. For this, we discuss notions of stability against both attacks and failures.At first, we investigate the computational complexity and approximability of finding resource-efficient attacks. This allows to point out limitations of an attacker's planning capabilities and demonstrates the influence of the chosen system parameters on the hardness of such attack problems.Then, we turn to study topology formation problems. Here, a set of topology parameters is given and the task consists in finding an eligible distribution topology. In particular, it has to minimize the maximum damage achievable by attacks with arbitrary attack parameters.We identify necessary and sufficient conditions on attack-stable distribution topologies. Thereby, we give mathematically sound guidelines for the topology management of peer-to-peer live streaming systems.We find large classes of efficiently-constructable topologies minimizing the system-wide packet loss under attacks. Additionally, we show that determining this feature for arbitrary topologies is coNP-complete.Considering topologies minimizing the maximum number of peers for which an attack leads to a heavy decrease in perceived streaming quality, the requirements change. Here, we show that the corresponding topology formation problem is closely related to long-standing open problems of Design and Coding Theory.Finally, we study topologies minimizing the expected packet loss due to uncoordinated peer failures. We investigate properties and existence conditions of such topologies. Furthermore, we determine the computational complexity of constructing them.Our results provide guidelines for the topology management of peer-to-peer live streaming systems and mathematically determine which goals can be achieved efficiently

The selection of networks of nature reserves.

Setting aside networks of protected areas for conservation is urgently needed to counteract the current extinction crisis. Complementarity-based reserve selection algorithms have been developed in recognition that such a task needs to make the best possible use of the scarce resources available to conservation, maximising the return in terms of biodiversity protection. This project aims to contribute to the improvement of these algorithms, particularly using optimisation methods, to make them more applicable to practical reserve selection. In pursuing this objective, a number of different approaches are adopted. Using different exemplar data sets, I (i) explore methods for the evaluation of existing networks of protected areas; (H) develop guidelines for the selection of networks which are more robust to species temporal turnover, and present evidence that minimum complementary sets tend to select areas of ecological transition; (Hi) demonstrate how optimisation tools can be applied to maximise phylogenetic diversity, and present evidence that complementary sets maximising for taxonomic richness are adequate surrogates in representing phylogenetic diversity; (iv) demonstrate how species rarity influences complementary reserve selection across geopolitical boundaries; (v) provide guidelines for the application of reserve selection algorithms in areas with poor biological data; and (vi) investigate what should be adequate conservation targets for reserve networks representing plant and vertebrate species, in the tropical rain forests and at a global scale. I then put the results obtained in this thesis and other published literature in a broader context, analysing the explanations as to why reserve selection algorithms are failing to have an impact in conservation practice. This study demonstrates the flexibility of reserve selection algorithms as tools for the selection of complementary reserve networks, and proposes developments needed to improve their effectiveness as practical conservation planning tools

Linear Programming Tools and Approximation Algorithms for Combinatorial Optimization

We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio. Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques. The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem. The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon