Computing optimal shortcuts for networks

Abstract

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.