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Günter Rote

Publications and source records attributed to Günter Rote.

At least 19 recordsLinked to original sources
arXiv2026

Finding a Shortest Curve that Separates Few Objects from Many

We present a fixed-parameter tractable (FPT) algorithm to find a shortest curve that separates some polygons from others. Formally, the input is a set of interior-disjoint simple polygons in the plane, where $k$ of the polygons are required to be enclosed and the remaining optional polygons have non-negative penalties. The goal is to find a closed curve that is disjoint from the polygon interiors and encloses the $k$ required polygons, while minimizing the length of the curve plus the penalties of the enclosed optional polygons. If the penalties are high, the output is a shortest curve that separates the required polygons from the others. The runtime of our algorithm is $O(3^k n^3)$, where $n$ is the number of vertices of the input polygons. The problem is NP-hard if $k$ is not fixed, even in very special cases. We extend the result to a graph version of the problem where the input is a connected plane graph with positive edge weights. There are $k$ required faces; the remaining faces are optional and have non-negative penalties. The goal is to find a closed walk in the graph that encloses the $k$ required faces, while minimizing the weight of the walk plus the penalties of the enclosed optional faces. We also consider an inverted version of the problem where the required objects must lie outside the curve. Our algorithms solve some other well-studied problems, such as geometric knapsack. Assuming the Exponential Time Hypothesis, we prove that neither the geometric nor the graph version of our problem can be solved in $2^{o(k)}\cdot n^{O(1)}$ time.

cs.CG↗
arXiv2026

A Gray code for arborescences of tournaments

We consider the following question of Knuth: given a directed graph $G$ and a root $r$, can the arborescences of $G$ rooted in $r$ be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of $G$ under arc flips, also called flip graph of $G$. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.

math.CO↗
arXiv2026

Compatible Triangulations of Simple Polygons

Let $P$ and $Q$ be simple polygons with $n$ vertices each. We wish to compute triangulations of $P$ and $Q$ that are combinatorially equivalent, if they exist. We consider two versions of the problem: if a triangulation of $P$ is given, we can decide in $O(n\log n + nr)$ time if $Q$ has a compatible triangulation, where $r$ is the number of reflex vertices of $Q$. If we are already given the correspondence between vertices of $P$ and $Q$ (but no triangulation), we can find compatible triangulations of $P$ and $Q$ in time $O(M(n))$, where $M(n)$ is the running time for multiplying two $n\times n$ matrices.

cs.CG↗
arXiv2025

Sweeping $x$-monotone pseudolines

We study the problem of sweeping a pseudoline arrangement with $n$ $x$-monotone curves with a rope (an $x$-monotone curve that connects the points at infinity). The rope can move by flipping over a face of the arrangement, replacing parts of it from the lower to the upper chain of the face. Counting as length of the rope the number of edges, what rope-length can be needed in such a sweep? We show that all such arrangements can be swept with rope-length at most $2n-2$, and for some arrangements rope-length at least $7(n-2)/4+1$ is required. We also discuss some complexity issues around the problem of computing a sweep with the shortest rope-length.

cs.CG↗
arXiv2025

On Solving Simple Curved Nonograms

Nonograms are a popular type of puzzle, where an arrangement of curves in the plane (in the classic version, a rectangular grid) is given together with a series of hints, indicating which cells of the subdivision are to be colored. The colored cells yield an image. Curved nonograms use a curve arrangement rather than a grid, leading to a closer approximation of an arbitrary solution image. While there is a considerable amount of previous work on the natural question of the hardness of solving a classic nonogram, research on curved nonograms has so far focused on their creation, which is already highly non-trivial. We address this gap by providing algorithmic and hardness results for curved nonograms of varying complexity.

cs.CG↗
arXiv2025

NumPSLA -- An experimental research tool for pseudoline arrangements and order types

We present a program for enumerating all pseudoline arrangements with a small number of pseudolines and abstract order types of small point sets. This program supports computer experiments with these structures, and it complements the order-type database of Aichholzer, Aurenhammer, and Krasser. This system makes it practical to explore the abstract order types for 12 points, and the pseudoline arrangements of 11 pseudolines.

math.CO↗
arXiv2025

Minimum spanning blob-trees

We investigate blob-trees, a new way of connecting a set of points, by a mixture of enclosing them by cycles (as in the convex hull) and connecting them by edges (as in a spanning tree). We show that a minimum-cost blob-tree for $n$ points can be computed in $O(n^3)$ time.

cs.CG↗
arXiv2024

Probabilistic Finite Automaton Emptiness is Undecidable for a Fixed Automaton

We construct a probabilistic finite automaton (PFA) with 7 states and an input alphabet of 5 symbols for which the PFA Emptiness Problem is undecidable. The only input for the decision problem is the starting distribution. For the proof, we use reductions from special instances of the Post Correspondence Problem. We also consider some variations: The input alphabet of the PFA can be restricted to a binary alphabet at the expense of a larger number of states. If we allow a rational output value for each state instead of a yes-no acceptance decision, the number of states can even be reduced to 6.

cs.FL↗
arXiv2024

Probabilistic Finite Automaton Emptiness is undecidable

It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present three proofs of this theorem from the literature in a self-contained way, and we derive some strengthenings. For example, we show that the problem remains undecidable for a fixed probabilistic finite automaton with 11 states, where only the starting distribution is given as input.

cs.FL↗
arXiv2024

Grid Peeling of Parabolas

Grid peeling is the process of repeatedly removing the convex hull vertices of the grid-points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.

cs.CG↗
arXiv2023

Removing Popular Faces in Curve Arrangements

A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces.

cs.CG↗
arXiv2022

Towards a Geometric Understanding of the 4-Dimensional Point Groups

We classify the finite groups of orthogonal transformations in 4-space, and we study these groups from the viewpoint of their geometric action, using polar orbit polytopes. For one type of groups (the toroidal groups), we develop a new classification based on their action on an invariant torus, while we rely on classic results for the remaining groups. As a tool, we develop a convenient parameterization of the oriented great circles on the 3-sphere, which leads to (oriented) Hopf fibrations in a natural way.

math.MG↗
arXiv2021

Optimal strategies in fractional games: vertex cover and domination

In a hypergraph with vertex set $V$ and edge set $E$, a real-valued function $f: V \to [0, 1]$ is a fractional transversal if $\sum_{v\in e} f(v) \ge 1$ for every edge $e \in E$. Its size is $|f| := \sum_{v \in V} f(v)$, and the fractional transversal number is the smallest possible $|f|$. We consider a game scenario where two players with opposite goals construct a fractional transversal incrementally, trying to minimize and maximize $|f|$, respectively. We prove that both players have strategies to achieve their common optimum, and they can reach their goals using rational weights.

math.CO↗
arXiv2020

An almost optimal bound on the number of intersections of two simple polygons

What is the maximum number of intersections of the boundaries of a simple $m$-gon and a simple $n$-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of $m$ and $n$ is even: If both $m$ and $n$ are even, then every pair of sides may cross and so the answer is $mn$. If exactly one polygon, say the $n$-gon, has an odd number of sides, it can intersect each side of the $m$-gon at most $n-1$ times; hence there are at most $mn-m$ intersections. It is not hard to construct examples that meet these bounds. If both $m$ and $n$ are odd, the best known construction has $mn-(m+n)+3$ intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only $mn-(m + \lceil \frac{n}{6} \rceil)$, for $m \ge n$. We prove a new upper bound of $mn-(m+n)+C$ for some constant $C$, which is optimal apart from the value of $C$.

math.CO↗
arXiv2019

PL Morse theory in low dimensions

We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular we show that in low dimensions $d \leq 4$ a homologically regular point on a PL $d$-manifold is always strongly regular. Examples show that this fails to hold in higher dimensions $d \geq 5$. One of our constructions involves an 8-vertex embedding of the dunce hat into a polytopal 4-sphere with 8 vertices such that a regular neighborhood is Mazur's contractible 4-manifold.

math.GT↗
arXiv2019

FPT Algorithms for Diverse Collections of Hitting Sets

In this work, we study the $d$-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of $r$ solutions of size at most $k$ each, which has recently been introduced to the field of parameterized complexity [Baste et al., 2019]. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process which models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are FPT in $k + r$ for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of `base' solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.

cs.DS↗