arXiv subjects

Daniel Bertschinger

Publications and source records attributed to Daniel Bertschinger.

8 recordsLinked to original sources

arXiv2026

Bowties and Hourglasses: Intersections of Double-Wedges (or Stabbing and Avoiding Line Segments)

We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be either bowties (which do not contain a vertical line) or hourglasses (which contain a vertical line), in contrast to earlier studies that focused on arrangements of only bowties. This generalization changes the setting drastically, in particular, with respect to all arguments involving the point-line duality. Namely, a point in the intersection of all double-wedges is equivalent to a line that stabs a set of segments $\mathcal{S}$ (corresponding to the bowties) while it avoids a different set of segments $\mathcal{A}$ (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of $n$ double-wedges may consist of $\Omega(n^2)$ interior-disjoint regions. Further, we discuss Gallai-type results for arrangements of segments and anti-segments, and we provide algorithms for computing the intersection of such arrangements with worst-case optimal running time. Finally, we also prove that we can find a single intersection point in almost optimal running time, assuming that 3SUM admits no truly subquadratic-time algorithm.

cs.CG↗

arXiv2023

The Complexity of Recognizing Geometric Hypergraphs

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is associated with a point $p_v\in \mathbb{R}^d$ and each hyperedge $e\in E$ is associated with a connected set $s_e\subset \mathbb{R}^d$ such that $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$ for all $e\in E$. We say that a given hypergraph $H$ is representable by some (infinite) family $F$ of sets in $\mathbb{R}^d$, if there exist $P\subset \mathbb{R}^d$ and $S \subseteq F$ such that $(P,S)$ is a geometric representation of $H$. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is $\exists\mathbb{R}$-hard for halfspaces in $\mathbb{R}^d$. We study the families of translates of balls and ellipsoids in $\mathbb{R}^d$, as well as of other convex sets, and show that their RECOGNITION problems are also $\exists\mathbb{R}$-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

cs.CG↗

arXiv2022

Well-Separation and Hyperplane Transversals in High Dimensions

A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that $k$ point sets are well-separated if and only if their convex hulls admit no $(k - 2)$-transversal, i.e., if there exists no $(k - 2)$-dimensional flat that intersects the convex hulls of all $k$ sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a $(d - 1)$-transversal) of a family of $d + 1$ line segments in $\mathbb{R}^d$, where $d$ is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an $\Omega\left(\frac{\log k}{k \log \log k}\right)$-approximation algorithm that is polynomial in $d$ and $k$, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a $(k - 2)$-transversal is in fact strongly NP-complete.

cs.CG↗

arXiv2022

Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\exists\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\mathsf{NP}=\exists\mathbb{R}$.

cs.CC↗

arXiv2021

Topological Art in Simple Galleries

Let $P$ be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in $P$. We say two points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is contained in $P$. We denote by $V(P)$ the family of all minimum guard placements. The Hausdorff distance makes $V(P)$ a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy equivalent to $S$. Furthermore, for various concrete topological spaces $T$, we describe instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to $T$.

cs.CG↗

arXiv2021

Tukey Depth Histograms

The Tukey depth of a flat with respect to a point set is a concept that appears in many areas of discrete and computational geometry. In particular, the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges can all be phrased in terms of depths of certain flats with respect to one or more point sets. In this work, we introduce the Tukey depth histogram of $k$-flats in $\mathbb{R}^d$ with respect to a point set $P$, which is a vector $D^{k,d}(P)$, whose $i$'th entry $D^{k,d}_i(P)$ denotes the number of $k$-flats spanned by $k+1$ points of $P$ that have Tukey depth $i$ with respect to $P$. As our main result, we give a complete characterization of the depth histograms of points, that is, for any dimension $d$ we give a description of all possible histograms $D^{0,d}(P)$. This then allows us to compute the exact number of possible such histograms.

cs.CG↗

arXiv2020

Weighted Epsilon-Nets

Motivated by recent work of Bukh and Nivasch on one-sided $\varepsilon$-approximants, we introduce the notion of \emph{weighted $\varepsilon$-nets}. It is a geometric notion of approximation for point sets in $\mathbb{R}^d$ similar to $\varepsilon$-nets and $\varepsilon$-approximations, where it is stronger than the former and weaker than the latter. The main idea is that small sets can contain many points, whereas large sets must contain many points of the weighted $\varepsilon$-net. In this paper, we analyze weak weighted $\varepsilon$-nets with respect to convex sets and axis-parallel boxes and give upper and lower bounds on $\varepsilon$ for weighted $\varepsilon$-nets of size two and three. Some of these bounds apply to classical $\varepsilon$-nets as well.

cs.CG↗

arXiv2020

An Optimal Decentralized $(\Delta + 1)$-Coloring Algorithm

Consider the following simple coloring algorithm for a graph on $n$ vertices. Each vertex chooses a color from $\{1, \dotsc, \Delta(G) + 1\}$ uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected $O(n \log \Delta)$ steps, which is optimal and proves a conjecture of Chakrabarty and Supinski [SOSA'20].

cs.DS↗