ﻻ يوجد ملخص باللغة العربية
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 $Theta(log chi (G))$, while for 3D the minimum scan time is not upper bounded by $chi (G)$. 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 $kleq chi(G)^c$. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph.
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
A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions neede
A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d in {2,3}$) is calle
We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(alpha_0,ldots, alpha_{n-1})$, $alpha_iin (-pi,pi)$, for $iin{0,ldots, n-1}$. The problem of realizing $A$ by a polygon ca
In this paper, we consider the Minimum-Load $k$-Clustering/Facility Location (MLkC) problem where we are given a set $P$ of $n$ points in a metric space that we have to cluster and an integer $k$ that denotes the number of clusters. Additionally, we