arXiv subjects

Benedikt Hahn

Publications and source records attributed to Benedikt Hahn.

4 recordsLinked to original sources

arXiv2026

Blow-ups of order types of positive density

Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $\kappa$-colored sequence of $n \ge d+1$ points in general position in $\mathbb{R}^d$. Let $\rho$ be a $\kappa$-colored order type on $k \le d+1$ points that has positive density on $P$; that is, for some constant $\delta >0$, there are $\delta \cdot \binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $\rho$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, \delta$, $k$ and $\kappa$) and disjoint subsets $X_1,\dots,X_k$ of $P$, each with at least $c \cdot n$ points, such that for every choice of $k$ points $x_i \in X_i$, $(x_1,\dots,x_k)$ has the same order type and color pattern as $\rho$.

math.CO↗

arXiv2026

Integrating Causal Machine Learning into Clinical Decision Support Systems: Insights from Literature and Practice

Current clinical decision support systems (CDSSs) typically base their predictions on correlation, not causation. In recent years, causal machine learning (ML) has emerged as a promising way to improve decision-making with CDSSs by offering interpretable, treatment-specific reasoning. However, existing research often emphasizes model development rather than designing clinician-facing interfaces. To address this gap, we investigated how CDSSs based on causal ML should be designed to effectively support collaborative clinical decision-making. Using a design science research methodology, we conducted a structured literature review and interviewed experienced physicians. From these, we derived eight empirically grounded design requirements, developed seven design principles, and proposed nine practical design features. Our results establish guidance for designing CDSSs that deliver causal insights, integrate seamlessly into clinical workflows, and support trust, usability, and human-AI collaboration. We also reveal tensions around automation, responsibility, and regulation, highlighting the need for an adaptive certification process for ML-based medical products.

cs.HC↗

arXiv2025

Edge densities of drawings of graphs with one forbidden cell

A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell $c$ is the cyclic sequence of crossings and vertices along the boundary walk of $c$. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an $n$-vertex multigraph $G$ does not contain any such triangular cell, Ackerman and Tardos [JCTA 2007] proved that $G$ has at most $8n-20$ edges, while Kaufmann, Klemz, Knorr, Reddy, Schr\"oder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of $\mathfrak{c}$-free drawings, that is, drawings that do not contain any cell of one fixed cell type $\mathfrak{c}$, and investigate the edge density of the corresponding graphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible cell type $\mathfrak{c}$. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type $\mathfrak{c}$ that the edge density of $n$-vertex (multi)graphs with $\mathfrak{c}$-free drawings is either linear in $n$ or superlinear in $n$. In most cases, our bounds are tight up to an additive constant. We further consider the question which simple graphs admit a simple drawing without some given cell type(s). For the class of cell types that are not incident to any crossing, we give a complete characterization of all simple graphs that admit a simple drawing without any such cell. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from $7n-28$ to $7.5n-28$.

math.CO↗

arXiv2025

On the geometric $k$-colored crossing number of $K_n$

We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We substantially improve asymptotic upper bounds on $\overline{\overline{\text{cr}}}_k(K_n)$ for $k=2,\ldots, 10$ by developing a procedure for general $k$ that derives $k$-edge colored drawings of $K_n$ for arbitrarily large $n$ from initial drawings with a low number of monochromatic crossings. We obtain the latter by heuristic search, employing a \textsc{MAX-$k$-CUT}-formulation of a subproblem in the process.

cs.CG↗