Point Sweep Coverage on Path

TL;DR

This paper investigates the computational complexity of deploying mobile sensors to periodically cover points of interest on a line, providing NP-hardness proofs and approximation algorithms for various sensor velocity scenarios.

Contribution

It proves NP-Completeness for the Point Sweep Coverage problem on a line with diverse sensor velocities and offers polynomial-time and approximation algorithms for maximizing coverage.

Findings

NP-Completeness of the decision problem for different sensor velocities.

Polynomial-time optimal solution for uniform velocities in weighted coverage.

Approximation algorithms with ratios of 1/2 and 1/2(1-1/e) for general velocity cases.

Abstract

An important application of wireless sensor networks is the deployment of mobile sensors to periodically monitor (cover) a set of points of interest (PoIs). The problem of Point Sweep Coverage is to deploy fewest sensors to periodically cover the set of PoIs. For PoIs in a Eulerian graph, this problem is known NP-Hard even if all sensors are with uniform velocity. In this paper, we study the problem when PoIs are on a line and prove that the decision version of the problem is NP-Complete if the sensors are with different velocities. We first formulate the problem of Max-PoI sweep coverage on path (MPSCP) to find the maximum number of PoIs covered by a given set of sensors, and then show it is NP-Hard. We also extend it to the weighted case, Max-Weight sweep coverage on path (MWSCP) problem to maximum the sum of the weight of PoIs covered. For sensors with uniform velocity, we give a polynomial-time optimal solution to MWSCP. For sensors with constant kinds of velocities, we present a $\frac{1}{2}$-approximation algorithm. For the general case of arbitrary velocities, we propose two algorithms. One is a $\frac{1}{2\alpha}$-approximation algorithm family scheme, where integer $\alpha\ge2$ is the tradeoff factor to balance the time complexity and approximation ratio. The other is a $\frac{1}{2}(1-1/e)$-approximation algorithm by randomized analysis.