On the complexity of and algorithms for min-max target coverage on a line boundary

Abstract

Given a set of sensors distributed on the plane and a set of Point of Interests (POIs) on a line segment, a primary task of the mobile wireless sensor network is to schedule a coverage of the POIs by the sensors, such that each POI is monitored by at least one sensor. For balancing the energy consumption, we study the min-max line barrier target coverage (LBTC) problem which aims to minimize the maximum movement of the sensors from their original positions to final positions for the coverage. We first proved that when the radius of the sensors are non-uniform integers, even 1-dimensional LBTC (1D-LBTC), a special case of LBTC in which the sensors are distributed on the line segment instead of the plane, is NP -hard. The hardness result is interesting, since the continuous version of LBTC of covering a given line segment instead of the POIs is known polynomial solvable [2]. Then we presented an exact algorithm for LBTC with sensors of uniform radius distributed on the plane, via solving the decision version of LBTC. We showed that our algorithm always finds an optimal solution in time O(mn(logm+logn)> to LBTC when there exists any, where m and n are the numbers of POIs and sensors.