Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts

We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network. We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves. Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices

Computing optimal shortcuts for networks

We augment a plane Euclidean network with a segment or shortcut to minimize the largest distance between any two points along the edges of the resulting network. In this continuous setting, the problem of computing distances and placing a shortcut is much harder as all points on the network, instead of only the vertices, must be taken into account. Our main result for general networks states that it is always possible to determine in polynomial time whether the network has an optimal shortcut and compute one in case of existence. We also improve this general method for networks that are paths, restricted to using two types of shortcuts: those of any fixed direction and shortcuts that intersect the path only on its endpoints.Peer ReviewedPostprint (published version

Computing optimal shortcuts for networks

We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts.Peer ReviewedPostprint (published version

Augmenting Graphs to Minimize the Radius

We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time (5/3-?)-approximation algorithm, for any ? > 0, unless P = NP. We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth

Computing Optimal Shortcuts for Networks

We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus N of a plane Euclidean network N by in- serting iteratively a finite set of segments, called shortcut set , while reducing the diameterof the locus of the resulting network. There are two main differences with the classicalaugmentation problems: the endpoints of the segments are allowed to be points of N as well as points of the previously inserted segments (instead of only vertices of N ), and the notion of diameter is adapted to the fact that we deal with N instead of N . This increases enormously the hardness of the problem but also its possible practical applications to net- work design. Among other results, we characterize the existence of shortcut sets, computethem in polynomial time, and analyze the role of the convex hull of N when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of ashortcut set is NP-hard, one can always determine in polynomial time whether insertingonly one segment suffices to reduce the diameter.Ministerio de Economía y Competitividad MTM2015-63791-