Placing your coins on a shelf

We consider the problem of packing a family of disks ``on a shelf,'' that is, such that each disk touches the x -axis from above and such that no two disks overlap. We study the problem of minimizing the distance between the leftmost point and the rightmost point of any disk in such a packing.  We show how to approximate this problem within a factor of 4/3 in O(n log n) time. We further provide an O(n log n) -time exact algorithm for a special case which includes inputs where the ratio between the largest radius and the smallest radius is less than four.  On the negative side, we prove that the problem is NP-hard even when the ratio between the largest radius and the smallest radius is at most 36.