Minimum Scan Cover with Angular Transition Costs

03/19/2020
by   Sandor P. Fekete, et al.
0

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 Θ(logχ (G)), while for 3D the minimum scan time is not upper bounded by χ (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 3/2, even for bipartite graphs; conversely, we present a 9/2-approximation algorithm for this scenario. Generally, we give an O(c)-approximation for k-colored graphs with k≤χ(G)^c. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 2

03/26/2021

Minimum Scan Cover and Variants – Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by c...
04/24/2019

Reoptimization of Path Vertex Cover Problem

Most optimization problems are notoriously hard. Considerable efforts mu...
01/11/2022

Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

Eternal Vertex Cover problem is a dynamic variant of the vertex cover pr...
07/25/2018

Mildly Exponential Time Approximation Algorithms for Vertex Cover, Uniform Sparsest Cut and Related Problems

In this work, we study the trade-off between the running time of approxi...
10/30/2020

Monitoring the edges of a graph using distances

We introduce a new graph-theoretic concept in the area of network monito...
03/09/2018

Geometric and LP-based heuristics for the quadratic travelling salesman problem

A generalization of the classical TSP is the so-called quadratic travell...
04/30/2018

Decoupling Respiratory and Angular Variation in Rotational X-ray Scans Using a Prior Bilinear Model

Data-driven respiratory signal extraction from rotational X-ray scans ha...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.