  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.
