Minimum k-Connected Geometric Networks

Keywords and Synonyms

Geometric graphs; Euclidean graphs          

Problem Definition

The following classical optimization problem is considered: for a given undirected weighted geometric network, find its minimum-cost sub-network that satisfies a priori given multi-connectivity requirements.

Notations

Let ${ G = (V,E) }$ be a  geometric network, whose vertex set V corresponds to a set of n points in ℝ ^d for certain integer d, ${ d \ge 2 }$, and whose edge set E corresponds to a set of straight-line segments connecting pairs of points in V. G is called complete if E connects all pairs of points in V.

The cost ${ \delta(x,y) }$ of an edge connecting a pair of points ${ x, y \in \mathbb{R}^d }$ is equal to the Euclidean distance between points x and y, that is, ${ \delta(x,y) = \sqrt{\sum_{i=1}^d (x_i - y_i)^2} }$, where ${ x = (x_1, \dots, x_d) }$ and ${ y = (y_1, \dots, y_d) }$. More generally, the cost ${ \delta(x,y) }$ could be defined using other norms, such as \...