In this paper, we consider a problem of covering a straight line segment by equal circles that are initially arbitrarily placed on a plane by moving their centers on a segment or on a straight line containing a segment so that the segment is completely covered, the neighboring circles in the cover are touching each other and the total length of the paths traveled by circles is minimal. The complexity status of the problem is not known. We propose a $O(n^{2+c}/\varepsilon^2)$--time FPTAS for this problem, where $n$ is the number of circles and $c>0$ is arbitrarily small real.