1. Connectivity with uncertainty regions given as line segmentsSergio Cabello, David Gajser, 2024, original scientific article Abstract: For a set ${\mathcal Q}$ of points in the plane and a real number $\delta \ge 0$, let $\mathbb{G}_\delta({\mathcal Q})$ be the graph defined on ${\mathcal Q}$ by connecting each pair of points at distance at most $\delta$. We consider the connectivity of $\mathbb{G}_\delta({\mathcal Q})$ in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set ${\mathcal P}$ of $n-k$ points in the plane and a set ${\mathcal S}$ of $k$ line segments in the plane, find the minimum $\delta \ge 0$ with the property that we can select one point $p_s\in s$ for each segment $s\in {\mathcal S}$ and the corresponding graph $\mathbb{G}_\delta( {\mathcal P}\cup \{ p_s\mid s\in {\mathcal S}\})$ is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in ${\mathcal O}(f(k) n \log n)$ time, for a computable function $f(\cdot)$. This implies that the problem is FPT when parameterized by $k$. The best previous algorithm uses ${\mathcal O}((k!)^k k^{k+1}\cdot n^{2k})$ time and computes the solution up to fixed precision.
Keywords: computational geometry, uncertainty, geometric optimization, fixed parameter tractability, parametric search
Published in DiRROS: 13.05.2024; Views: 49; Downloads: 24
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2. Maximum matchings in geometric intersection graphsÉdouard Bonnet, Sergio Cabello, Wolfgang Mulzer, 2023, original scientific article Abstract: Let $G$ be an intersection graph of $n$ geometric objects in the plane. We show that a maximum matching in $G$ can be found in $O(\rho^{3\omega/2}n^{\omega/2})$ time with high probability, where $\rho$ is the density of the geometric objects and $\omega>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^\omega)$ time. The same result holds for any subgraph of $G$, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in $O(n^{\omega/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, \Psi]$ can be found in $O(\Psi^6\log^{11} n + \Psi^{12 \omega} n^{\omega/2})$ time with high probability.
Keywords: computational geometry, geometric intersection graphs, disk graphs, unit-disk graphs, matchings
Published in DiRROS: 08.04.2024; Views: 115; Downloads: 45
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3. Faster distance-based representative skyline and k-center along pareto front in the planeSergio Cabello, 2023, original scientific article Abstract: We consider the problem of computing the distance-based representative skyline in the plane, a problem introduced by Tao, Ding, Lin and Pei and independently considered by Dupin, Nielsen and Talbi in the context of multi-objective optimization. Given a set $P$ of $n$ points in the plane and a parameter $k$, the task is to select $k$ points of the skyline defined by $P$ (also known as Pareto front for $P$) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in $O(n \log h)$ time, where $h$ is the number of points in the skyline of $P$. We also show that the decision problem can be solved in $O(n \log k)$ time and the optimization problem can be solved in $O(n \log k + n \log\log n)$ time. This improves previous algorithms and is optimal for a large range of values of $k$.
Keywords: geometric optimization, skyline, pareto front, clustering, k-center
Published in DiRROS: 15.03.2024; Views: 124; Downloads: 56
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