Before revision:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively. There also exist two algorithms with approximation factor more than 2, for the c-packedness using axis-aligned hypercubes instead of disks, where they consider the shapes centered at the vertices of the curve when determining the c-packedness. We call this measure the vertex-relative c-packedness. For axis-aligned squares instead of disks, a related concept called relative-length was already discussed and it has an exact{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^3) algorithm.
After revision:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n (\log^2 n)(\log^2 \frac{1{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^3) algorithm.