Fine-grained dichotomies for the Tutte plane and Boolean #CSP

Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $y=1$, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given $n$-vertex graph cannot be determined in time $exp(o(n))$ unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with $n$ vertices cannot be computed in time $exp(o(n))$ unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.Comment: 16 pages, 1 figur

Paths and walks, forests and planes : arcadian algorithms and complexity

This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und Komplexitätstheorie. Wir entwickeln eine neue Technik für schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender Graßmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener Länge in gerichteten Graphen schneller als bisher zu schätzen. Außerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener Länge zu finden, falls der Eingabegraph nur wenige solche Pfade enthält. Übrigens entwickeln wir schnellere deterministische Algorithmen, um Spannbäume mit wenigen Blättern zu finden. Wir studieren außerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-Komplexität davon, die genaue Anzahl von Wäldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollständigen wir die Komplexitätsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass Wälder mit gegebener Kantenzahl zu zählen, wenigstens so schwer ist, wie Cliquen gegebener Größe zu zählen.Cluster of Excellence (Multimodal Computing and Interaction

Fixed-Parameter Algorithms for Computing RAC Drawings of Graphs

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends (or determining that none exists) is fixed-parameter tractable parameterized by either the feedback edge number of $G$, or $b$ plus the vertex cover number of $G$.Comment: Accepted at GD 202

A Note on the Approximability of Deepest-Descent Circuit Steps

Linear programs (LPs) can be solved by polynomially many moves along the circuit direction improving the objective the most, so-called deepest-descent steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv, 2019), a consequence of the hardness of deciding the existence of an optimal circuit-neighbor (OCNP) on LPs with non-unique optima. We prove OCNP is easy under the promise of unique optima, but already $O(n^{1-\varepsilon})$-approximating dd-steps remains hard even for totally unimodular $n$-dimensional 0/1-LPs with a unique optimum. We provide a matching $n$-approximation

Accelerated Degradation Protocols for Iridium-Based Oxygen Evolving Catalysts in Water Splitting Devices

Hydrogen production by proton exchange membrane (PEM) water electrolysis is among the promising energy storage solutions to buffer an increasingly volatile power grid employing significant amounts of renewable energies. In PEM electrolysis research, 24 h galvanostatic measurements are the most common initial stability screenings and up to 5,000 h are used to assess extended stability, while commercial stack runtimes are within the 20,000–50,000 h range. In order to obtain stability data representative of commercial lifetimes with significantly reduced test duration an accelerated degradation test (ADT) was suggested by our group earlier. Here, we present a study on the broad applicability of the suggested ADT in RDE and CCM measurements and showcase the advantage of transient over static operation for enhanced catalyst degradation studies. The suggested ADT-1.6 V protocol allows unprecedented, reproducible and quick assessment of anode catalyst long-term stability, which will strongly enhance degradation research and reliability. Furthermore, this protocol allows to bridge the gap between more fundamental RDE and commercially relevant CCM studies

Early blood glucose profile and neurodevelopmental outcome at two years in neonatal hypoxic-ischaemic encephalopathy

Background: To examine the blood glucose profile and the relationship between blood glucose levels and neurodevelopmental outcome in term infants with hypoxic-ischaemic encephalopathy. Methods: Blood glucose values within 72 hours of birth were collected from 52 term infants with hypoxic-ischaemic encephalopathy. Hypoglycaemia [ 150 mg/dL (8.3 mmol/L)] were correlated to neurodevelopmental outcome at 24 months of age. Results: Four fifths of the 468 blood samples were in the normoglycaemic range (392/468:83.8%). Of the remaining 76 samples, 51.3% were in the hypoglycaemic range and (48.7%) were hyperglycaemic. A quarter of the hypoglycaemic samples (28.2%:11/39) and a third of the hyperglycaemic samples (32.4%:12/37) were recorded within the first 30 minutes of life. Mean (SD) blood glucose values did not differ between infants with normal and abnormal outcomes [4.89(2.28) mmol/L and 5.02(2.35) mmol/L, p value = 0.15] respectively. In term infants with hypoxic-ischaemic encephalopathy, early hypoglycaemia (between 0-6 hours of life) was associated with adverse outcome at 24 months of age [OR = 5.8, CI = 1.04-32)]. On multivariate analysis to adjust for grade of HIE this association was not statistically significant. Late hypoglycaemia (6-72 hours of life) was not associated with abnormal outcome [OR = 0.22, CI (0.04-1.14)]. The occurrence of hyperglycaemia was not associated with adverse outcome. Conclusion: During the first 72 hours of life, blood glucose profile in infants with hypoxic-ischaemic encephalopathy varies widely despite a management protocol. Early hypoglycaemia (0-6 hours of life) was associated with severe HIE, and thereby; adverse outcome