Hardness of Minimum Barrier Shrinkage and Minimum Installation Path

19 Aug 2020
•
Cabello Sergio
•
de Verdière Éric Colin

In the Minimum Installation Path problem, we are given a graph $G$ with edge
weights $w(. )$ and two vertices $s,t$ of $G$...We want to assign a non-negative
power $p(v)$ to each vertex $v$ of $G$ so that the edges $uv$ such that
$p(u)+p(v)$ is at least $w(uv)$ contain some $s$-$t$-path, and minimize the sum
of assigned powers. In the Minimum Barrier Shrinkage problem, we are given a
family of disks in the plane and two points $x$ and $y$ lying outside the
disks. 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.(read more)