Euclidean Capacitated Facility Location Problem
If I rmemeber correctly, you have to approximate the number of clients connected to each gate. Otherwise, you would immediately get something like $O(n^{O(g)})$, where $g$ is the number of gates in a subproblem. By approximating this number up to a facotr of $(1+\varepsilon/\log n)$ throughout the dynamic programming one can get a $(1+\varepsilon)$ error in the end. That would yield running times similar to what you stated above.