Sebastian Brandt

On consistent migration of flows in SDNs

We study consistent migration of flows, with special focus on software defined networks. Given a current and a desired network flow configuration, we give the first polynomial-time algorithm to decide if a congestion-free migration is possible. However, if all flows must be integer or are unsplittable, this is NP-hard to decide. A similar problem is providing increased bandwidth to an application, while keeping all other flows in the network, but possibly migrating them consistently to other paths. We show that the maximum increase can be approximated arbitrarily well in polynomial time. Current methods as RSVP-TE consider unsplittable flows and remove flows of lesser importance in order to increase bandwidth for an application: We prove that deciding what flows need to be removed is an NP-hard optimization problem with no PTAS possible unless P = NP.

A lower bound for the distributed Lovász local lemma

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].

Collaboration Without Communication: Evacuating Two Robots from a Disk

We consider the problem of evacuating two robots from a bounded area, through an unknown exit located on the boundary. Initially, the robots are in the center of the area and throughout the evacuation process they can only communicate with each other when they are at the same point at the same time. Having a visibility range of 0, the robots can only identify the location of the exit if they are already at the exit position. The task is to minimize the time it takes until both robots reach the exit, for a worst-case placement of the exit. For unit disks, an upper bound of 5.628 for the evacuation time is presented in [8]. Using the insight that, perhaps surprisingly, a forced meeting of the two robots as performed in the respective algorithm does not provide an exchange of any non-trivial information, we design a simpler algorithm that achieves an upper bound of 5.625. Our numerical simulations suggest that this bound is optimal for the considered natural class of algorithms. For dealing with the technical difficulties in analyzing the algorithm, we formulate a powerful new criterion that, for a given algorithm, reduces the number of possible worst-case exits radically. This criterion is of independent interest and can be applied to any area shape. Due to space restrictions, this version of the paper contains no proofs or illustrating figures; the full version can be found at http://disco.ethz.ch/publications/ciac2017-robotevac.pdf.

Augmenting anycast network flows

Updating network flows in a real-world setting is a nascent research area, especially with the recent rise of Software Defined Networks. While augmenting s-t flows of a single commodity is a well-understood concept, we study updating flows in a multi-commodity setting: Given a directed network with flows of different commodities, how can the capacity of some commodities be increased, without reducing capacities of other commodities, when moving flows in the network in an orchestrated order? To this extent, we show how the notion of augmenting flows can be efficiently extended to multiple commodities for anycast applications.

LCL Problems on Grids

LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log* n), or Θ(n), and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log* n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log* n) or Θ(n) in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is Θ(log* n), we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form A o Sk, where A is a finite function, Sk is an algorithm for finding a maximal independent set in kth power of the grid, and k is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.

Augmenting flows for the consistent migration of multi-commodity single-destination flows in SDNs

Abstract Updating network flows in a real-world setting is a nascent research area, especially with the recent rise of Software Defined Networks. While augmenting s - t flows of a single commodity is a well-understood concept, we study updating flows in a multi-commodity setting: Given a directed network with flows of different commodities, how can the capacity of some commodities be increased, without reducing capacities of other commodities, when moving flows in the network in an orchestrated order? To this extent, we show how the notion of augmenting flows can be efficiently extended to multiple commodities for applications with a single logical destination. We also show that our methods induce stronger consistency settings than previous work. Lastly, we prove the consistent migration to new demands to be NP-hard for unsplittable flows, and discuss extensions for the case of multiple source–destination pairs.

A Tight Lower Bound for the Capture Time of the Cops and Robbers Game.

For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When k >= 2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(n^{k + 1}), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(n^{k + 1}) bound is asymptotically tight for any constant k >= 2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.

Kategorisierung von Röntgenbildern mit aktiven Konturmodellen

Fur das Image Retrieval in Medical Applications (IRMA) mussen digitalen Radiographien automatisch Korperregionen zugeordnet werden. Experimentell werden Methoden zum formbasierten Image Retrieval auf radiologische Bilder angewendet. Es wird untersucht, ob die Umrislinie dargestellter Korperteile mit einem Ballon-Modell aufgefunden werden kann. Anschliesend werden semilokale invariante Signaturen ermittelt und in ihren Klassifikationseigenschaften mit invarianten Momenten und Fourier-Koeffizienten verglichen. Eine dem visuellen Eindruck entsprechende Konturfindung mit dem Ballon-Modell gelingt auf 496 von 1616 Radiographien, die Nearest-Neighbour-Klassifikation auf Basis der extrahierten Formmerkmale in sechs Kategorien erreicht bisher nur Klassifikationsraten von 65%.

Fine-grained Lower Bounds on Cops and Robbers

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniels conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time.nIn recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game.

Approximating Small Balanced Vertex Separators in Almost Linear Time

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any