Hardness of Minimum Barrier Shrinkage and Minimum Installation Path

In the MINIMUM INSTALLATION PATH problem, we are given a graph G with edge weights w(center dot) and two vertices s, t of G. We want to assign a non-negative power p: V -> R->= 0 to the vertices of G, so that the activated edges {uv is an element of E(G) vertical bar p(u) + p(v) >= w(uv)} contain some s-t-path, and minimize the sum of assigned powers. In the MINIMUM BARRIER SHRINKAGE problem, we are given, in the plane, a family of disks and two points x and y. The task is to shrink the disks, each one possibly by a different amount, so that we can draw an x-y curve that is disjoint from the interior of the shrunken disks, and the sum of the decreases in the radii is minimized. We show that the MINIMUM INSTALLATION PATH and the MINIMUM BARRIER SHRINKAGE problems (or, more precisely, the natural decision problems associated with them) are weakly NP-hard. (c) 2020 Elsevier B.V. All rights reserved.