Minimum Scan Cover with Angular Transition Costs

by   Sandor P. Fekete, et al.

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.


Minimum Scan Cover and Variants – Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by c...

Reoptimization of Path Vertex Cover Problem

Most optimization problems are notoriously hard. Considerable efforts mu...

Approximation algorithms for hitting subgraphs

Let H be a fixed undirected graph on k vertices. The H-hitting set probl...

Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

Eternal Vertex Cover problem is a dynamic variant of the vertex cover pr...

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...

Toward a Dichotomy for Approximation of H-coloring

The minimum cost homomorphism problem (MinHOM) is a natural optimization...

Star Routing: Between Vehicle Routing and Vertex Cover

We consider an optimization problem posed by an actual newspaper company...

Please sign up or login with your details

Forgot password? Click here to reset