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.Comment: 14 pages, 3 figure

New bounds on the average distance from the Fermat-Weber center of a planar convex body

The Fermat-Weber center of a planar body $Q$ is a point in the plane from which the average distance to the points in $Q$ is minimal. We first show that for any convex body $Q$ in the plane, the average distance from the Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot \Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490 \cdot \Delta(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3} \cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber center of a planar convex body, in Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878, Springer, pp. 132-14

Minimum Weight Euclidean (1+ε)(1+\varepsilon)-Spanners

Given a set $S$ of $n$ points in the plane and a parameter $\varepsilon>0$, a Euclidean $(1+\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that contains, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)\|pq\|$. We show that the minimum weight of a Euclidean $(1+\varepsilon)$-spanner for $n$ points in the unit square $[0,1]^2$ is $O(\varepsilon^{-3/2}\,\sqrt{n})$, and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline $O(\varepsilon^{-2}\sqrt{n})$, obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on $n$ points in $[0,1]^2$, and a tight bound for the lightness of Euclidean $(1+\varepsilon)$-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to Euclidean $d$-space for every dimension $d\in \mathbb{N}$: The minimum weight of a Euclidean $(1+\varepsilon)$-spanner for $n$ points in the unit cube $[0,1]^d$ is $O_d(\varepsilon^{(1-d^2)/d}n^{(d-1)/d})$, and this bound is the best possible. For the $n\times n$ section of the integer lattice, we show that the minimum weight of a Euclidean $(1+\varepsilon)$-spanner is between $\Omega(\varepsilon^{-3/4}\cdot n^2)$ and $O(\varepsilon^{-1}\log(\varepsilon^{-1})\cdot n^2)$. These bounds become $\Omega(\varepsilon^{-3/4}\cdot \sqrt{n})$ and $O(\varepsilon^{-1}\log(\varepsilon^{-1})\cdot \sqrt{n})$ when scaled to a grid of $n$ points in the unit square. In particular, this shows that the integer grid is \emph{not} an extremal configuration for minimum weight Euclidean $(1+\varepsilon)$-spanners.Comment: 27 pages, 9 figures. An extended abstract appears in the Proceedings of WG 202

On RAC Drawings of Graphs with Two Bends per Edge

It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.Comment: Presented at the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve