Hardness of Minimum Barrier Shrinkage and Minimum Installation Path

TL;DR

This paper proves that the problems of Minimum Installation Path and Minimum Barrier Shrinkage are weakly NP-hard, highlighting their computational difficulty in graph and geometric settings respectively.

Contribution

The paper establishes the weak NP-hardness of two new problems: Minimum Installation Path and Minimum Barrier Shrinkage, expanding understanding of their computational complexity.

Findings

Both problems are weakly NP-hard.

The results apply to graph and geometric scenarios.

Complexity implications for related optimization problems.

Abstract

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.