On the Complexity of Barrier Resilience for Fat Regions

TL;DR

This paper investigates the computational complexity of the barrier resilience problem with fat obstacle regions, proving NP-hardness in general but providing fixed-parameter tractable algorithms and approximation schemes for specific cases.

Contribution

It establishes NP-hardness for fat regions with bounded ply and develops FPT algorithms and approximation methods for unit disks and similar fat regions.

Findings

NP-hardness for fat regions with bounded ply

FPT algorithms for unit disks and similar fat regions

An $(1+ ext{epsilon})$-approximation algorithm with specific runtime

Abstract

In the \emph {barrier resilience} problem (introduced by Kumar {\em et al.}, Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, we show that the problem is NP-hard when the collection only contains fat regions with bounded ply $\Delta$ (even when they are axis-aligned rectangles of aspect ratio $1 : (1 + \varepsilon)$). We also show that the problem is fixed-parameter tractable (FPT) for unit disks and for similarly-sized $\beta$-fat regions with bounded ply $\Delta$ and $O(1)$ pairwise boundary intersections. Furthermore, we can use our FPT algorithm to construct an $(1+\varepsilon)$-approximation algorithm that runs in $O(2^{f(\Delta, \varepsilon,\beta)}n^5)$ time, where $f\in O(\frac{\Delta^4\beta^8}{\varepsilon^4}\log(\beta\Delta/\varepsilon))$.