“Green” barrier coverage with mobile sensors

Mobile sensors are located on a barrier represented by a line segment. Each sensor has a single energy source that can be used for both moving and sensing. A sensor consumes energy in movement in proportion to distance traveled, and it expends energy per time unit for sensing in direct proportion to its radius raised to a constant exponent. We address the problem of energy efficient coverage. The input consists of the initial locations of the sensors and a coverage time requirement t. A feasible solution consists of an assignment of destinations and coverage radii to all sensors such that the barrier is covered. We consider two variants of the problem that are distinguished by whether the radii are given as part of the input. In the fixed radii case, we are also given a radii vector ρ, and the radii assignment r must satisfy r_i ∈ {0, ρ_i}, for every i, while in the variable radii case the radii assignment is unrestricted. We consider two objective functions. In the first the goal is to minimize the sum of the energy spent by all sensors and in the second the goal is to minimize the maximum energy used by any sensor. We present FPTASs for the problem of minimizing the energy sum with variable radii and for the problem of minimizing the maximum energy with variable radii. We also show that the latter can be approximated within any additive constant ε > 0. We show that the problem of minimizing the energy sum with fixed radii cannot be approximated within a factor of O(n^c), for any constant c, unless P=NP. The problem of minimizing the maximum energy with fixed radii is shown to be strongly NP-hard. Additional results are given for three special cases: (i) sensors are stationary, (ii) free movement, and (iii) uniform fixed radii.