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We show that Not-All-Equal 3-Sat remains NP-complete when restricted to instances that simultaneously satisfy the following properties: (i) The clauses are given as the disjoint union of k partitions, for any fixed $k \geq 4$, of the variable set into subsets of size 3, and (ii) each pair of distinct clauses shares at most one variable. Property (i) implies that each variable appears in exactly $k$ clauses and each clause consists of exactly 3 unnegated variables. Therewith, we improve upon our earlier result (Darmann and D\"ocker, 2020). Complementing the hardness result for at least $4$ partitions, we show that for $k\leq 3$ the corresponding decision problem is in P. In particular, for $k\in \{1,2\}$, all instances that satisfy Property (i) are nae-satisfiable. By the well-known correspondence between Not-All-Equal 3-Sat and hypergraph coloring, we obtain the following corollary of our results: For $k\geq 4$, Bicolorability is NP-complete for linear 3-uniform $k$-regular hypergraphs even if the edges are given as a decomposition into $k$ perfect matchings; with the same restrictions, for $k \leq 3$ Bicolorability is in P, and for $k \in \{1,2\}$ all such hypergraphs are bicolorable. Finally, we deduce from a construction in the work by Pilz (Pilz, 2019) that every instance of Positive Planar Not-All-Equal Sat with at least three distinct variables per clause is nae-satisfiable. Hence, when restricted to instances with a planar incidence graph, each of the above variants of Not-All-Equal 3-Sat turns into a trivial decision problem.

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bl\"asius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from $(2/\sqrt{3})$ to $4/3$ depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

Rooted phylogenetic networks are used by biologists to infer and represent complex evolutionary relationships between species that cannot be accurately explained by a phylogenetic tree. Tree-child networks are a particular class of rooted phylogenetic networks that has been extensively investigated in recent years. In this paper, we give a novel characterisation of a tree-child network $\mathcal{R}$ in terms of cherry-picking sequences that are sequences on the leaves of $\mathcal{R}$ and reduce it to a single vertex by repeatedly applying one of two reductions to its leaves. We show that our characterisation extends to unrooted tree-child networks which are mostly unexplored in the literature and, in turn, also offers a new approach to settling the computational complexity of deciding if an unrooted phylogenetic network can be oriented as a rooted tree-child network.

The aim of this paper is to provide a coherent framework for transforming boundary pairs of digital images from one resolution to another without knowledge of the full images. It is intended to facilitate the simultaneous usage of multiresolution processing and boundary reduction, primarily for algorithms in computational dynamics and computational control theory.

This work stages Foosball as a versatile platform for advancing scientific research, particularly in the realm of robot learning. We present an automated Foosball table along with its corresponding simulated counterpart, showcasing a diverse range of challenges through example tasks within the Foosball environment. Initial findings are shared using a simple baseline approach. Foosball constitutes a versatile learning environment with the potential to yield cutting-edge research in various fields of artificial intelligence and machine learning, notably robust learning, while also extending its applicability to industrial robotics and automation setups. To transform our physical Foosball table into a research-friendly system, we augmented it with a 2 degrees of freedom kinematic chain to control the goalkeeper rod as an initial setup with the intention to be extended to the full game as soon as possible. Our experiments reveal that a realistic simulation is essential for mastering complex robotic tasks, yet translating these accomplishments to the real system remains challenging, often accompanied by a performance decline. This emphasizes the critical importance of research in this direction. In this concern, we spotlight the automated Foosball table as an invaluable tool, possessing numerous desirable attributes, to serve as a demanding learning environment for advancing robotics and automation research., Comment: 7 pages, 5 figures, 2024 IEEE International Conference on Robotics and Automation (ICRA 2024)

In comparison to phylogenetic trees, phylogenetic networks are more suitable to represent complex evolutionary histories of species whose past includes reticulation such as hybridisation or lateral gene transfer. However, the reconstruction of phylogenetic networks remains challenging and computationally expensive due to their intricate structural properties. For example, the small parsimony problem that is solvable in polynomial time for phylogenetic trees, becomes NP-hard on phylogenetic networks under softwired and parental parsimony, even for a single binary character and structurally constrained networks. To calculate the parsimony score of a phylogenetic network $N$, these two parsimony notions consider different exponential-size sets of phylogenetic trees that can be extracted from $N$ and infer the minimum parsimony score over all trees in the set. In this paper, we ask: What is the maximum difference between the parsimony score of any phylogenetic tree that is contained in the set of considered trees and a phylogenetic tree whose parsimony score equates to the parsimony score of $N$? Given a gap-free sequence alignment of multi-state characters and a rooted binary level-$k$ phylogenetic network, we use the novel concept of an informative blob to show that this difference is bounded by $k+1$ times the softwired parsimony score of $N$. In particular, the difference is independent of the alignment length and the number of character states. We show that an analogous bound can be obtained for the softwired parsimony score of semi-directed networks, while under parental parsimony on the other hand, such a bound does not hold.

We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.

Today we rely on networks that are created and maintained by smart devices. For such networks, there is no governing central authority but instead the network structure is shaped by the decisions of selfish intelligent agents. A key property of such communication networks is that they should be easy to navigate for routing data. For this, a common approach is greedy routing, where every device simply routes data to a neighbor that is closer to the respective destination. Networks of intelligent agents can be analyzed via a game-theoretic approach and in the last decades many variants of network creation games have been proposed and analyzed. In this paper we present the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in our model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled. Besides this, the agents optimize their connection quality within the created network by aiming for greedy routing paths with low stretch. For our model, we analyze the existence of (approximate)-equilibria and the computational hardness in different underlying metric spaces. E.g., we characterize the set of equilibria in 1-2-metrics and tree metrics and show that Nash equilibria always exist. For Euclidean space, the setting which is most relevant in practice, we prove that equilibria are not guaranteed to exist but that the well-known $\Theta$-graph construction yields networks having a low stretch that are game-theoretically almost stable. For general metric spaces, we show that approximate equilibria exist where the approximation factor depends on the cost of maintaining any link.

Context. Acoustic waves in the Sun are affected by the atmospheric layers, but this region is often ignored in forward models due to the increase in computational cost. Aims. The purpose of this work is to take into account the solar atmosphere without increasing significantly the computational cost. Methods. We solve a scalar wave equation that describes the propagation of acoustic modes inside the Sun using a finite element method. The boundary conditions used to truncate the computational domain are learned from the Dirichlet-to-Neumann operator, that is the relation between the solution and its normal derivative at the computational boundary. These boundary conditions may be applied at any height above which the background medium is assumed to be radially symmetric. Results. Taking into account the atmosphere is important even for wave frequencies below the acoustic cut-off. In particular, the mode frequencies computed for an isothermal atmosphere differ by up 10 {\mu}Hz from those computed for the VAL-C atmospheric model. We show that learned infinite elements lead to a numerical accuracy similar to that obtained for a traditional radiation boundary condition. Its main advantage is to reproduce the solution for any radially symmetric atmosphere to a very good accuracy at a low computational cost. Conclusions. This work emphasizes the importance of including atmospheric layers in helioseismology and proposes a computationally efficient method to do so.

Recently, there has been a growing interest in the relationships between unrooted and rooted phylogenetic networks. In this context, a natural question to ask is if an unrooted phylogenetic network U can be oriented as a rooted phylogenetic network such that the latter satisfies certain structural properties. In a recent preprint, Bulteau et al. claim that it is computational hard to decide if U has a funneled (resp. funneled tree-child) orientation, for when the internal vertices of U have degree at most 5. Unfortunately, the proof of their funneled tree-child result appears to be incorrect. In this paper, we present a corrected proof and show that hardness remains for other popular classes of rooted phylogenetic networks such as funneled normal and funneled reticulation-visible. Additionally, our results hold regardless of whether U is rooted at an existing vertex or by subdividing an edge with the root.

Wearable devices continuously collect sensor data and use it to infer an individual's behavior, such as sleep, physical activity, and emotions. Despite the significant interest and advancements in this field, modeling multimodal sensor data in real-world environments is still challenging due to low data quality and limited data annotations. In this work, we investigate representation learning for imputing missing wearable data and compare it with state-of-the-art statistical approaches. We investigate the performance of the transformer model on 10 physiological and behavioral signals with different masking ratios. Our results show that transformers outperform baselines for missing data imputation of signals that change more frequently, but not for monotonic signals. We further investigate the impact of imputation strategies and masking rations on downstream classification tasks. Our study provides insights for the design and development of masking-based self-supervised learning tasks and advocates the adoption of hybrid-based imputation strategies to address the challenge of missing data in wearable devices., Comment: Paper accepted in Human-Centric Representation Learning workshop at AAAI 2024 (https://hcrl-workshop.github.io/2024/)

The gradient-flow formalism proves to be a useful tool in lattice calculations of quantum chromodynamics. For example, it can be used as a scheme to renormalize composite operators by inverting the short-flow-time expansion of the corresponding flowed operators. In this paper, we consider the short-flow-time expansion of five quark bilinear operators, the scalar, pseudoscalar, vector, axialvector, and tensor currents, and compute the matching coefficients through next-to-next-to-leading order QCD. Among other applications, our results constitute one ingredient for calculating bag parameters of mesons within the gradient-flow formalism on the lattice., Comment: v2: matches published version

Cosine similarity is an established similarity metric for computing associations on vectors, and it is commonly used to identify related samples from biological perturbational data. The distribution of cosine similarity changes with the covariance of the data, and this in turn affects the statistical power to identify related signals. The relationship between the mean and covariance of the distribution of the data and the distribution of cosine similarity is poorly understood. In this work, we derive the asymptotic moments of cosine similarity as a function of the data and identify the criteria of the data covariance matrix that minimize the variance of cosine similarity. We find that the variance of cosine similarity is minimized when the eigenvalues of the covariance matrix are equal for centered data. One immediate application of this work is characterizing the null distribution of cosine similarity over a dataset with non-zero covariance structure. Furthermore, this result can be used to optimize over a set of transformations or representations on a dataset to maximize power, recall, or other discriminative metrics, with direct application to noisy biological data. While we consider the specific biological domain of perturbational data analysis, our result has potential application for any use of cosine similarity or Pearson's correlation on data with covariance structure., Comment: 30 pages, 4 figures

Decentralized Federated Learning (DFL) emerges as an innovative paradigm to train collaborative models, addressing the single point of failure limitation. However, the security and trustworthiness of FL and DFL are compromised by poisoning attacks, negatively impacting its performance. Existing defense mechanisms have been designed for centralized FL and they do not adequately exploit the particularities of DFL. Thus, this work introduces Sentinel, a defense strategy to counteract poisoning attacks in DFL. Sentinel leverages the accessibility of local data and defines a three-step aggregation protocol consisting of similarity filtering, bootstrap validation, and normalization to safeguard against malicious model updates. Sentinel has been evaluated with diverse datasets and data distributions. Besides, various poisoning attack types and threat levels have been verified. The results improve the state-of-the-art performance against both untargeted and targeted poisoning attacks when data follows an IID (Independent and Identically Distributed) configuration. Besides, under non-IID configuration, it is analyzed how performance degrades both for Sentinel and other state-of-the-art robust aggregation methods.

Finding similar patients is a common objective in precision medicine, facilitating treatment outcome assessment and clinical decision support. Choosing widely-available patient features and appropriate mathematical methods for similarity calculations is crucial. International Statistical Classification of Diseases and Related Health Problems (ICD) codes are used worldwide to encode diseases and are available for nearly all patients. Aggregated as sets consisting of primary and secondary diagnoses they can display a degree of comorbidity and reveal comorbidity patterns. It is possible to compute the similarity of patients based on their ICD codes by using semantic similarity algorithms. These algorithms have been traditionally evaluated using a single-term expert rated data set. However, real-word patient data often display varying degrees of documented comorbidities that might impair algorithm performance. To account for this, we present a scale term that considers documented comorbidity-variance. In this work, we compared the performance of 80 combinations of established algorithms in terms of semantic similarity based on ICD-code sets. The sets have been extracted from patients with a C25.X (pancreatic cancer) primary diagnosis and provide a variety of different combinations of ICD-codes. Using our scale term we yielded the best results with a combination of level-based information content, Leacock & Chodorow concept similarity and bipartite graph matching for the set similarities reaching a correlation of 0.75 with our expert's ground truth. Our results highlight the importance of accounting for comorbidity variance while demonstrating how well current semantic similarity algorithms perform., Comment: 11 pages, 6 figures, 1 table

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber., Comment: 39 pages, 9 figures

In this paper we study the threshold model of \emph{geometric inhomogeneous random graphs} (GIRGs); a generative random graph model that is closely related to \emph{hyperbolic random graphs} (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their \emph{connectivity}, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a \emph{giant} component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability $1 - \exp(-\Omega(n^{(3-\tau)/2}))$ for graph size $n$ and a degree distribution with power-law exponent $\tau \in (2, 3)$. Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.

High-bandwidth signals are needed in many applications like radar, sensing, measurement and communications. Especially in optical networks, the sampling rate and analog bandwidth of digital-to-analog converters (DACs) is a bottleneck for further increasing data rates. To circumvent the sampling rate and bandwidth problem of electronic DACs, we demonstrate the generation of wide-band signals with low-bandwidth electronics. This generation is based on orthogonal sampling with sinc-pulse sequences in N parallel branches. The method not only reduces the sampling rate and bandwidth, at the same time the effective number of bits (ENOB) is improved, dramatically reducing the requirements on the electronic signal processing. In proof of concept experiments the generation of analog signals, as well as Nyquist shaped and normal data will be shown. In simulations we investigate the performance of 60 GHz data generation by 20 and 12 GHz electronics. The method can easily be integrated together with already existing electronic DAC designs and would be of great interest for all high-bandwidth applications.