# Approximation Algorithms for Minimizing Maximum Sensor Movement for Line Barrier Coverage in the Plane

## TL;DR

This paper introduces three approximation algorithms for the MMSM problem in line barrier coverage, improving maximum movement bounds and providing practical solutions with near-optimal guarantees.

## Contribution

It presents three novel approximation algorithms for MMSM, utilizing greedy, LP rounding, and matching techniques, with improved bounds and efficiency.

## Key findings

- The greedy algorithm achieves a maximum movement of D* + 2r_max.
- LP rounding and matching algorithms improve the bound to D* + r_max.
- A factor-2 approximation algorithm enhances performance when r_max > D*.

## Abstract

Given a line barrier and a set of mobile sensors distributed in the plane, the Minimizing Maximum Sensor Movement problem (MMSM) for \textcolor{black}{line barrier coverage} is to compute relocation positions for the sensors in the plane such that the barrier is entirely covered by the monitoring area of the sensors while the maximum relocation movement (distance) is minimized. Its weaker version, decision MMSM is to determine whether the barrier can be covered by the sensors within a given relocation distance bound $D\in\mathbb{Z}^{+}$. This paper presents three approximation algorithms for decision MMSM. The first is a simple greedy approach, which runs in time $O(n\log n)$ and achieves a maximum movement $D^{*}+2r_{max}$, where $n$ is the number of the sensors, $D^{*}$ is the maximum movement of an optimal solution and $r_{max}$ is the maximum radii of the sensors. The second and the third algorithms improve the maximum movement to $D^{*}+r_{max}$ , running in time $O(n^{7}L)$ and $O(R^{2}\sqrt{\frac{M}{\log R}})$ by applying linear programming (LP) rounding and maximal matching tchniques respecitvely, where $R=\sum2r_{i}$, which is $O(n)$ in practical scenarios of uniform sensing radius for all sensors, and $M\leq n\max r_{i}$. Applying the above algorithms for $O(\log(d_{max}))$ time in binary search immediately yields solutions to MMSM with the same performance guarantee. In addition, we also give a factor-2 approximation algorithm which can be used to improve the performance of the first three algorithms when $r_{max}>D^{*}$. As shown in \cite{dobrev2015complexity}, the 2-D MMSM problem admits no FPTAS as it is strongly NP-complete, so our algorithms arguably achieve the best possible ratio.