  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.
