Approximating Barrier Resilience for Arrangements of Non-identical Disk Sensors

Abstract

Let $\mathcal{A}$ be an arrangement of n sensors constituting a barrier between two regions S and T. The resilience of $\mathcal{A}$ with respect to S and T, denoted $\rho (\mathcal{A},S,T)$, is defined as the number of sensors in $\mathcal{A}$ that must be removed in order for there to be an S − T path that is not detected by any sensor. We introduce the Multi-Path Algorithm (MPA) and show that it guarantees a 2-approximation of $\rho (\mathcal{A},S,T)$ in time polynomial in n when sensors are unit disks in a two dimensional plane; this tightens to a 1.5-approximation when S and T are moderately well-separated. We also define a generalization of $\rho (\mathcal{A},S,T)$ denoted $\rho_{c} (\mathcal{A},S,T)$, which is the resilience of the barrier if regions covered by more than c distinct sensors in the original arrangement are treated as inaccessible. Then when the unit size constraint is relaxed, we prove that the MPA can still guarantee a 2-approximation of $\rho_{c} (\mathcal{A},S,T)$ for any constant c in time polynomial in n for arrangements of approximately equal-sized sensors.