Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus $\mathcal{N}_{\ell}$ of a plane Euclidean network $\mathcal{N}$ by inserting iteratively a finite set of segments, called \emph{shortcut set}, while reducing the diameter of the locus of the resulting network. There are two main differences with the classical augmentation problems: the endpoints of the segments are allowed to be points of $\mathcal{N}_{\ell}$ as well as points of the previously inserted segments (instead of only vertices of $\mathcal{N}$), and the notion of diameter is adapted to the fact that we deal with $\mathcal{N}_{\ell}$ instead of $\mathcal{N}$. This increases enormously the hardness of the problem but also its possible practical applications to network design. Among other results, we characterize the existence of shortcut sets, compute them in polynomial time, and analyze the role of the convex hull of $\mathcal{N}_{\ell}$ when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of a shortcut set is NP-hard, one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter.