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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: 37; Downloads: 17
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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: 116; Downloads: 54
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Keywords: automated performance prediction, autoML, single-objective black-box optimization, zero-shot learning
Published in DiRROS: 13.11.2023; Views: 345; Downloads: 207
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Keywords: black-box optimization, algorithms, problem instances, machine learning
Published in DiRROS: 15.09.2023; Views: 299; Downloads: 199
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Keywords: black-box single-objective optimization, optimization problem classification, problem representation, meta-learning
Published in DiRROS: 30.08.2023; Views: 337; Downloads: 231
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