Before revision:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
After revision:  There also exist two algorithms with approximation factor more than 2, for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.