arXiv subjects

Linda Kleist

Publications and source records attributed to Linda Kleist.

At least 37 records · Page 2Linked to original sources

arXiv2024

Online Sorting and Translational Packing of Convex Polygons

We investigate several online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container, while the aim is to minimize the used space. Among other variants, we consider strip packing and bin packing, where the container is the infinite horizontal strip $[0,\infty)\times [0,1]$ or a collection of $1 \times 1$ bins, respectively. We draw interesting connections to the following online sorting problem OnlineSorting$[γ,n]$: We receive a stream of real numbers $s_1,\ldots,s_n$, $s_i\in[0,1]$, one by one. Each real must be placed in an array $A$ with $γn$ initially empty cells without knowing the subsequent reals. The goal is to minimize the sum of differences of consecutive reals in $A$. The offline optimum is to place the reals in sorted order so the cost is at most $1$. We show that for any $Δ$-competitive online algorithm of OnlineSorting$[γ,n]$, it holds that $γΔ\inΩ(\log n/\log \log n)$. We use this lower bound to prove the non-existence of competitive algorithms for various online translational packing problems of convex polygons, among them strip packing, bin packing and perimeter packing. This also implies that there exists no online algorithm that can pack all streams of pieces of diameter and total area at most $δ$ into the unit square. These results are in contrast to the case when the pieces are restricted to rectangles, for which competitive algorithms are known. Likewise, the offline versions of packing convex polygons have constant factor approximation algorithms. As a complement, we also include algorithms for both online sorting and translation-only online strip packing with non-trivial competitive ratios. Our algorithm for strip packing relies on a new technique for recursively subdividing the strip into parallelograms of varying height, thickness and slope.

cs.CG↗

arXiv2023

A solution to Ringel's circle problem

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel's circle problem (1959). The proof relies on a (multidimensional) version of Gallai's theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.

math.CO↗

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↗

arXiv2023

Bounding the tripartite-circle crossing number of complete tripartite graphs

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. We present upper and lower bounds on the minimum number of crossings in tripartite-circle drawings of $K_{m,n,p}$. In contrast to 1- and 2-circle drawings, which may attain the Harary-Hill bound, our results imply that balanced restricted 3-circle drawings of the complete graph are not optimal.

math.CO↗

arXiv2023

The tripartite-circle crossing number of graphs with two small partition classes

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of $K_{a,b,n}$, where $a,b\leq 2$.

math.CO↗

arXiv2023

Adjacency Graphs of Polyhedral Surfaces

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $\mathbb{R}^3$. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $K_5$, $K_{5,81}$, or any nonplanar $3$-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $K_{4,4}$, and $K_{3,5}$ can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable $n$-vertex graphs is in $Ω(n \log n)$. From the non-realizability of $K_{5,81}$, we obtain that any realizable $n$-vertex graph has $O(n^{9/5})$ edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

cs.CG↗

arXiv2022

Competitive Location Problems: Balanced Facility Location and the One-Round Manhattan Voronoi Game

We study competitive location problems in a continuous setting, in which facilities have to be placed in a rectangular domain $R$ of normalized dimensions of $1$ and $ρ\geq 1$, and distances are measured according to the Manhattan metric. We show that the family of 'balanced' facility configurations (in which the Voronoi cells of individual facilities are equalized with respect to a number of geometric properties) is considerably richer in this metric than for Euclidean distances. Our main result considers the 'One-Round Voronoi Game' with Manhattan distances, in which first player White and then player Black each place $n$ points in $R$; each player scores the area for which one of its facilities is closer than the facilities of the opponent. We give a tight characterization: White has a winning strategy if and only if $ρ\geq n$; for all other cases, we present a winning strategy for Black.

cs.CG↗

arXiv2022

The Complexity of the Hausdorff Distance

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $\forall\exists_<\mathbb{R}$. This implies that the problem is NP-, co-NP-, $\exists\mathbb{R}$- and $\forall\mathbb{R}$-hard.

cs.CG↗

arXiv2022

Folding Polyiamonds into Octahedra

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid -- the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exist a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.

cs.CG↗

arXiv2022

Packing Squares into a Disk with Optimal Worst-Case Density

We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $δ=\frac{8}{5π}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\frac{8}{5}+\varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square $\left(\frac{1}{2}\right)$, circles in a square $\left(\fracπ{(3+2\sqrt{2})}\approx 0.539\right)$ and circles in a circle $\left(\frac{1}{2}\right)$ have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.

cs.CG↗

arXiv2021

Training Neural Networks is $\exists\mathbb R$-complete

Given a neural network, training data, and a threshold, it was known that it is NP-hard to find weights for the neural network such that the total error is below the threshold. We determine the algorithmic complexity of this fundamental problem precisely, by showing that it is $\exists\mathbb R$-complete. This means that the problem is equivalent, up to polynomial-time reductions, to deciding whether a system of polynomial equations and inequalities with integer coefficients and real unknowns has a solution. If, as widely expected, $\exists\mathbb R$ is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP. Neural networks are usually trained using some variation of backpropagation. The result of this paper offers an explanation why techniques commonly used to solve big instances of NP-complete problems seem not to be of use for this task. Examples of such techniques are SAT solvers, IP solvers, local search, dynamic programming, to name a few general ones.

cs.CC↗

arXiv2021

Scheduling with Machine Conflicts

We study the scheduling problem of makespan minimization while taking machine conflicts into account. Machine conflicts arise in various settings, e.g., shared resources for pre- and post-processing of tasks or spatial restrictions. In this context, each job has a blocking time before and after its processing time, i.e., three parameters. We seek for conflict-free schedules in which the blocking times of no two jobs intersect on conflicting machines. Given a set of jobs, a set of machines, and a graph representing machine conflicts, the problem SchedulingWithMachineConflicts (SMC), asks for a conflict-free schedule of minimum makespan. We show that, unless $\textrm{P}=\textrm{NP}$, SMC on $m$ machines does not allow for a $\mathcal{O}(m^{1-\varepsilon})$-approximation algorithm for any $\varepsilon>0$, even in the case of identical jobs and every choice of fixed positive parameters, including the unit case. Complementary, we provide approximation algorithms when a suitable collection of independent sets is given. Finally, we present polynomial time algorithms to solve the problem for the case of unit jobs on special graph classes. Most prominently, we solve it for bipartite graphs by using structural insights for conflict graphs of star forests.

cs.DM↗

arXiv2021

Completeness for the Complexity Class $\forall \exists \mathbb{R}$ and Area-Universality

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists \mathbb{R}$ plays a crucial role in the study of geometric problems. Sometimes $\exists \mathbb{R}$ is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $\exists \mathbb{R}$ deals with existentially quantified real variables. In analogy to $Π_2^p$ and $Σ_2^p$ in the famous polynomial hierarchy, we study the complexity classes $\forall \exists \mathbb{R}$ and $\exists \forall \mathbb{R}$ with real variables. Our main interest is the area-universality problem, where we are given a plane graph $G$, and ask if for each assignment of areas to the inner faces of $G$, there exists a straight-line drawing of $G$ realizing the assigned areas. We conjecture that area-universality is $\forall \exists \mathbb{R}$-complete and support this conjecture by proving $\exists \mathbb{R}$- and $\forall \exists \mathbb{R}$-completeness of two variants of area-universality. To this end, we introduce tools to prove $\forall \exists \mathbb{R}$-hardness and membership. Finally, we present geometric problems as candidates for $\forall \exists \mathbb{R}$-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.

cs.CG↗

arXiv2021

Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

We show that the decision problem of determining whether a given (abstract simplicial) $k$-complex has a geometric embedding in $\mathbb R^d$ is complete for the Existential Theory of the Reals for all $d\geq 3$ and $k\in\{d-1,d\}$. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution. Moreover, this implies NP-hardness and constitutes the first hardness results for the algorithmic problem of geometric embedding (abstract simplicial) complexes.

cs.CC↗

arXiv2021

Minimum Scan Cover and Variants -- Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

cs.CG↗

arXiv2020

Upward Point Set Embeddings of Paths and Trees

We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The given point set $S$ has size at least the number of vertices in the tree. For the special case where the tree is a path $P$ we show that: (a) If $S$ is one-sided convex, the number of UPSEs equals the number of maximal monotone paths in $P$. (b) If $S$ is in general position and $P$ is composed by three maximal monotone paths, where the middle path is longer than the other two, then it always admits an UPSE on $S$. We show that the decision problem of whether there exists an UPSE of a directed tree with $n$ vertices on a fixed point set $S$ of $n$ points is NP-complete, by relaxing the requirements of the previously known result which relied on the presence of cycles in the graph, but instead fixing position of a single vertex. Finally, by allowing extra points, we guarantee that each directed caterpillar on $n$ vertices and with $k$ switches in its backbone admits an UPSE on every set of $n 2^{k-2}$ points.

cs.CG↗

arXiv2020

Folding Polyominoes with Holes into a Cube

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special \emph{simple} holes guarantee foldability.

cs.CG↗

arXiv2020

Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (MSC), we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex takes some time proportional to the corresponding turn angle. Our goal is to minimize the time until all scans are completed, i.e., to compute a schedule of minimum makespan. We show that MSC is closely related to both graph coloring and the minimum (directed and undirected) cut cover problem; in particular, we show that the minimum scan time for instances in 1D and 2D lies in $Θ(\log χ(G))$, while for 3D the minimum scan time is not upper bounded by $χ(G)$. We use this relationship to prove that the existence of a constant-factor approximation implies $P=NP$, even for one-dimensional instances. In 2D, we show that it is NP-hard to approximate a minimum scan cover within less than a factor of $\frac{3}{2}$, even for bipartite graphs; conversely, we present a $\frac{9}{2}$-approximation algorithm for this scenario. Generally, we give an $O(c)$-approximation for $k$-colored graphs with $k\leq χ(G)^c$. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph.

cs.CG↗