On the Complexity of Barrier Resilience for Fat Regions

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 $\mathrm{\Delta }$ (even when they are axis-aligned rectangles of aspect ratio $1:(1+\epsilon )$). We also show that the problem is fixed-parameter tractable (FPT) for unit disks and for similarly-sized $\beta$-fat regions with bounded ply $\mathrm{\Delta }$ and $O(1)$ pairwise boundary intersections. Furthermore, we can use our FPT algorithm to construct an $(1+\epsilon )$-approximation algorithm that runs in $O(2^{f(\mathrm{\Delta },\epsilon ,\beta )}{n}^5)$ time, where $f\in O(\frac{\Delta^4\beta^8}{\varepsilon^4}\log(\beta\Delta/\varepsilon))$.