Constrained Clustering Problems and Parity Games

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. We study several clustering objectives. We begin with studying the Euclidean k-center problem. The k-center problem is a classical combinatorial optimization problem which asks to select k centers and assign each input point in a set P to one of the centers, such that the maximum distance of any input point to its assigned center is minimized. The Euclidean k-center problem assumes that the input set P is a subset of a Euclidean space and that each location in the Euclidean space can be chosen as a center. We focus on the special case with k = 1, the smallest enclosing ball problem: given a set of points in m-dimensional Euclidean space, find the smallest sphere enclosing all the points. We combine known results about convex optimization with structural properties of the smallest enclosing ball to create a new algorithm. We show that on instances with rational coefficients our new algorithm computes the exact center of the optimal solutions and has a worst-case run time that is polynomial in the size of the input. We use the new algorithm to show that we can solve the Euclidean k-center problem in polynomial time for constant k and dimension m. The general unconstrained clustering problems are mostly very well studied. The k-center problem for example allows for elegant 2-approximation algorithms(Gonzalez 1985, Hochbaum,Shmoys 1986). However, the situation becomes significantly more difficult when constraints are added to the problem. We first look at the fair clustering. The fairness constraint is motivated by the fact that the general process of computing a clustering may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation algorithm for the fair k-center problem and an O(t)-approximation algorithm for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation algorithm for fair k-center (Rösner, Schmidt 2018). We extend and improve the known results. Firstly, we give a 5-approximation algorithm for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximation algorithms for the fair version of all of the classical clustering objectives (k-center, k-supplier, k-median, k-means and facility location). The latter approximation algorithms are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where for example the centers are already fixed. The second clustering constraint we study is privacy: Here, we are asked to only open a center when at least l points will be assigned to it. We raise the question whether a general method can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that additionally respects privacy. We show how to combine privacy with several other constraints and obtain approximation algorithms for the k-center problem with several combinations of constraints. In this dissertation we also study parity games, a two player game played on a directed graph. We study the case in which one of the two players controls only a small number k of nodes and the other player controls the n-k other nodes of the game. Our main result is a fixed-parameter-tractable algorithm that solves bipartite parity games in time k^{O(sqrt{k})} O(n^3), and general parity games in time (p+k)^{O(sqrt{k})} O(pnm), where p is the number of distinct priorities and m is the number of edges. For all games with k = o(n) this improves the previously fastest algorithm by Jurdziński, Paterson, and Zwick (2008). We also obtain novel kernelization results and an improved deterministic algorithm for parity games on graphs with small average node-degree

Smoothed Analysis of the Successive Shortest Path Algorithm

The minimum-cost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudo-polynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. However, some of the algorithms' running times observed in empirical studies contrast the running times obtained by worst-case analysis not only in the order of magnitude but also in the ranking when compared to each other. For example, the Successive Shortest Path (SSP) algorithm, which has an exponential worst-case running time, seems to outperform the strongly polynomial Minimum-Mean Cycle Canceling algorithm. To explain this discrepancy, we study the SSP algorithm in the framework of smoothed analysis and establish a bound of $O(mn\phi)$ for the number of iterations, which implies a smoothed running time of $O(mn\phi (m + n\log n))$, where $n$ and $m$ denote the number of nodes and edges, respectively, and $\phi$ is a measure for the amount of random noise. This shows that worst-case instances for the SSP algorithm are not robust and unlikely to be encountered in practice. Furthermore, we prove a smoothed lower bound of $\Omega(m \phi \min\{n, \phi\})$ for the number of iterations of the SSP algorithm, showing that the upper bound cannot be improved for $\phi = \Omega(n)$.Comment: A preliminary version has been presented at SODA 201

Privacy Preserving Clustering with Constraints

The $k$-center problem is a classical combinatorial optimization problem which asks to find $k$ centers such that the maximum distance of any input point in a set $P$ to its assigned center is minimized. The problem allows for elegant $2$-approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least $\ell$ clients will be assigned to it. We show how to combine privacy with several other constraints

„Dikulsion“ im naturwissenschaftlichen Unterricht aktuelle Positionen und Routenplanung

Abels S, Stinken-Rösner L. „Dikulsion“ im naturwissenschaftlichen Unterricht aktuelle Positionen und Routenplanung. In: Watts E, Hoffmann C, eds. Digitale NAWIgation von Inklusion. Digitale Werkzeuge für einen inklusiven Naturwissenschaftsunterricht. Edition Fachdidaktiken. Wiesbaden: Springer VS; 2022: 5-20

Experimentiervideos im naturwissenschaftlichen Unterricht – Lehren und Lernen mit und durch VidEX

Meier M, Kastaun M, Stinken-Rösner L. Experimentiervideos im naturwissenschaftlichen Unterricht – Lehren und Lernen mit und durch VidEX. In: Watts E, Hoffmann C, eds. Digitale NAWIgation von Inklusion. Digitale Werkzeuge für einen inklusiven Naturwissenschaftsunterricht. Edition Fachdidaktiken. Wiesbaden: Springer VS; 2022: 51-65.Das Projekt VidEX – Lehren und Lernen mit Experimentiervideos erweitert das bisher existierende Angebot an digitalen Werkzeugen für den naturwissenschaftlichen Unterricht. Konzeptionelles Kernelement ist das Experimentiervideo mit Begleitmaterialien, welches weniger als Ersatz für das Realexperiment dienen soll, sondern neuartige, differenzierte Zugänge zum Erwerb von Kompetenzen aus dem Bereich der naturwissenschaftlichen Erkenntnisgewinnung bietet. Grundlegend für die Entwicklung und Gestaltung von Experimentiervideos sind die in den jeweiligen Curricula vorgegebenen zu erwerbenden Kompetenzen und Fachinhalte, Gestaltungsprinzipien aus der kognitionspsychologischen Forschung, Gestaltgesetze für Demonstrationsexperimente sowie Perspektiven von aktiven Lehrkräften zu Potenzialen in der unterrichtlichen Videonutzung und Einsatzszenarien. Die Kollaboration von Fachdidaktiker:innen und Lehrenden bei der Konzeption und Evaluation der Experimentiervideos ermöglicht es, passgenaue Angebote entlang individueller Lernvoraussetzungen diverser Lerngruppen zu gestalten, die insbesondere für einen inklusiven naturwissenschaftlichen Unterricht neue Zugänge und individuelles sowie kollaboratives Lernen ermöglichen. Werden Experimentiervideos bisher von Lehrkräften vornehmlich unter Vorbehalt und im Distanzunterricht eingesetzt, so geht die Vision von VidEX hierüber hinaus, indem immanente Barrieren im Prozess naturwissenschaftlicher Erkenntnisgewinnung durch den Einsatz videogestützter Lernmaterialien abgebaut und forschend angelegte Experimentierwege für alle Lernenden, im Sinne einer Scientific Literacy for all, ermöglicht werden sollen. Im Beitrag werden die konzeptionellen Leitgedanken und damit tragenden Elemente zu VidEX theoretisch angebunden und empirisch gestützt ausgeführt

Low neuronal metabolism during isoflurane-induced burst suppression is related to synaptic inhibition while neurovascular coupling and mitochondrial function remain intact

Deep anaesthesia may impair neuronal, vascular and mitochondrial function facilitating neurological complications, such as delirium and stroke. On the other hand, deep anaesthesia is performed for neuroprotection in critical brain diseases such as status epilepticus or traumatic brain injury. Since the commonly used anaesthetic propofol causes mitochondrial dysfunction, we investigated the impact of the alternative anaesthetic isoflurane on neuro-metabolism. In deeply anaesthetised Wistar rats (burst suppression pattern), we measured increased cortical tissue oxygen pressure (ptiO2), a 35% drop in regional cerebral blood flow (rCBF) and burst-associated neurovascular responses. In vitro, 3% isoflurane blocked synaptic transmission and impaired network oscillations, thereby decreasing the cerebral metabolic rate of oxygen (CMRO2). Concerning mitochondrial function, isoflurane induced a reductive shift in flavin adenine dinucleotide (FAD) and decreased stimulus-induced FAD transients as Ca2þ influx was reduced by 50%. Computer simulations based on experimental results predicted no direct effects of isoflurane on mitochondrial complexes or ATP-synthesis. We found that isoflurane-induced burst suppression is related to decreased ATP consumption due to inhibition of synaptic activity while neurovascular coupling and mitochondrial function remain intact. The neurometabolic profile of isoflurane thus appears to be superior to that of propofol which has been shown to impair the mitochondrial respiratory chai