arXiv subjects

Csaba D. Tóth

Publications and source records attributed to Csaba D. Tóth.

At least 73 records · Page 4Linked to original sources

arXiv2018

Online Unit Covering in Euclidean Space

We revisit the online Unit Covering problem in higher dimensions: Given a set of $n$ points in $\mathbb{R}^d$, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in $\mathbb{R}^d$ using Euclidean distance. The current best competitive ratio of an online algorithm, $O(2^d d \log{d})$, is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio $O(1.321^d)$, thereby sharply improving on the earlier record by a large exponential factor. In particular, the competitive ratios are $5$ for the plane and $12$ for $3$-space (the previous ratios were $7$ and $21$, respectively). For $d=3$, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual). (II) We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in $\mathbb{R}^d$ under the $L_{2}$ norm is at least $d+1$ for every $d \geq 1$. This greatly improves upon the previous best lower bound, $Ω(\log{d} / \log{\log{\log{d}}})$, due to Charikar et al. (2004). (III) We obtain lower bounds of $4$ and $5$ for the competitive ratio of any deterministic algorithm for online Unit Covering in $\mathbb{R}^2$ and respectively $\mathbb{R}^3$; the previous best lower bounds were both $3$. (IV) When the input points are taken from the square or hexagonal lattices in $\mathbb{R}^2$, we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of $3$.

cs.CG↗

arXiv2018

Crossing Minimization in Perturbed Drawings

Due to data compression or low resolution, nearby vertices and edges of a graph drawing may be bundled to a common node or arc. We model such a `compromised' drawing by a piecewise linear map $φ:G\rightarrow \mathbb{R}^2$. We wish to perturb $φ$ by an arbitrarily small $\varepsilon>0$ into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An $\varepsilon$-perturbation, for every $\varepsilon>0$, is given by a piecewise linear map $ψ_\varepsilon:G\rightarrow \mathbb{R}^2$ with $\|φ-ψ_\varepsilon\|<\varepsilon$, where $\|.\|$ is the uniform norm (i.e., $\sup$ norm). We present a polynomial-time solution for this optimization problem when $G$ is a cycle and the map $φ$ has no \emphh{spurs} (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when $G$ is an arbitrary graph and $φ$ has no spurs, and (ii) when $φ$ may have spurs and $G$ is a cycle or a union of disjoint paths.

cs.CG↗

arXiv2018

Maximum Area Axis-Aligned Square Packings

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\geq \frac12$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.

cs.CG↗

arXiv2017

Problems on Track Runners

Consider the circle $C$ of length 1 and a circular arc $A$ of length $\ell\in (0,1)$. It is shown that there exists $k=k(\ell) \in \mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds so that at any time $t \geq 0$, at least one of the $k$ runners is not in $A$. On the other hand, we show the following: Assume that $k$ runners $1,2,\ldots,k$, with constant rationally independent (thus distinct) speeds $ξ_1,ξ_2,\ldots,ξ_k$, run clockwise along a circle of length $1$, starting from arbitrary points. For every circular arc $A\subset C$ and for every $T>0$, there exists $t>T$ such that all runners are in $A$ at time $t$. Several other problems of a similar nature are investigated.

math.CO↗

arXiv2017

Recognizing Weakly Simple Polygons

We present an $O(n\log n)$-time algorithm that determines whether a given planar $n$-gon is weakly simple. This improves upon an $O(n^2\log n)$-time algorithm by Chang, Erickson, and Xu (2015). Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.

cs.CG↗

arXiv2017

Convex polygons in geometric triangulations

We show that the maximum number of convex polygons in a triangulation of $n$ points in the plane is $O(1.5029^n)$. This improves an earlier bound of $O(1.6181^n)$ established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of $Ω(1.5028^n)$ due to the same authors. Given a planar straight-line graph $G$ with $n$ vertices, we show how to compute efficiently the number of convex polygons in $G$.

math.MG↗

arXiv2017

Geometric Biplane Graphs I: Maximal Graphs

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.

cs.CG↗

arXiv2017

Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.

cs.CG↗

arXiv2016

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG $G=(V,E)$ in general position can be augmented to a 2-connected PSLG $(V,E\cup E^+)$ by adding new edges of total Euclidean length $\|E^+\|\leq 2\|E\|$, and this bound is the best possible. An optimal edge set $E^+$ can be computed in $O(|V|^4)$ time; however the problem becomes NP-hard when $G$ is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG $G=(V,E)$ into a planar straight-line cycle $G'=(V,E')$ such that $\|E'\|\leq 2\|{\rm MST}(V)\|$, and the graph remains connected with edge length below $\|E\|+\|{\rm MST}(V)\|$ at all stages. These bounds are the best possible.

cs.CG↗

arXiv2016

Note on k-planar crossing numbers

The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to the rectilinear variants of the $k$-planar crossing number.

math.CO↗

arXiv2016

Arc diagrams, flip distances, and Hamiltonian triangulations

We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity as a means to establish Hamiltonicity. But in general about $3n/5$ flips are necessary to reach a $4$-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on $n$ vertices from $5.2n-33.6$ to $5n-23$. We also show that for every triangulation on $n$ vertices there is a simultaneous flip of less than $2n/3$ edges to a $4$-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on $n$ vertices admits an arc diagram with less than $n/2$ biarcs, that is, after subdividing less than $n/2$ (of potentially $3n-6$) edges the resulting graph admits a $2$-page book embedding.

cs.CG↗

arXiv2016

Monotone Paths in Geometric Triangulations

(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $Ω(1.7003^n)$. (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the number of monotone paths in $G$ can be computed in $O(n^2)$ time.

cs.CG↗

arXiv2016

Reconstruction of weakly simple polygons from their edges

Given $n$ line segments in the plane, do they form the edge set of a \emph{weakly simple polygon}; that is, can the segment endpoints be perturbed by at most $\varepsilon$, for any $\varepsilon>0$, to obtain a simple polygon? While the analogous question for \emph{simple polygons} can easily be answered in $O(n\log n)$ time, we show that it is NP-complete for weakly simple polygons. We give $O(n)$-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are \emph{directed}, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the $n$ input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.

cs.CG↗

arXiv2016

Multi-Colored Spanning Graphs

We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given $n$ points in the plane, each labeled with one or more primary colors, a \emph{colored spanning graph} (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The \textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for $k$ primary colors when $k\ge 3$ and provide a $(2-\frac{1}{3+2\varrho})$-approximation algorithm for $k=3$ that runs in polynomial time, where $\varrho$ is the Steiner ratio. Further, we give a $O(n)$ time algorithm in the special case that the input points are collinear and $k$ is constant.

cs.CG↗

arXiv2016

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.

cs.CG↗

arXiv2016

The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of García et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.

cs.CG↗

arXiv2016

The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $Θ(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.

cs.DM↗

arXiv2016

Anchored Rectangle and Square Packings

For points $p_1,\ldots , p_n$ in the unit square $[0,1]^2$, an \emph{anchored rectangle packing} consists of interior-disjoint axis-aligned empty rectangles $r_1,\ldots , r_n\subseteq [0,1]^2$ such that point $p_i$ is a corner of the rectangle $r_i$ (that is, $r_i$ is \emph{anchored} at $p_i$) for $i=1,\ldots, n$. We show that for every set of $n$ points in $[0,1]^2$, there is an anchored rectangle packing of area at least $7/12-O(1/n)$, and for every $n\in \mathbf{N}$, there are point sets for which the area of every anchored rectangle packing is at most $2/3$. The maximum area of an anchored \emph{square} packing is always at least $5/32$ and sometimes at most $7/27$. The above constructive lower bounds immediately yield constant-factor approximations, of $7/12 -\varepsilon$ for rectangles and $5/32$ for squares, for computing anchored packings of maximum area in $O(n\log n)$ time. We prove that a simple greedy strategy achieves a $9/47$-approximation for anchored square packings, and $1/3$ for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in $n^{O(1/\varepsilon)}$ and $\exp({\rm poly}(\log (n/\varepsilon)))$ time, respectively.

cs.CG↗