I believe you are seeking the optimal TSP disk tour, for congruent disks,
a special case of "TSP with neighborhoods."
The TSP disk-tour problem is NP-hard,
but there exists a PTAS (polynomial-time approximation scheme) for it.
See the paper by Dumitrescu & Mitchell below.
Their algorithm achieves an approximation ratio of $1 + 8/\pi \approx 3.55$,
and is very efficient (although it would not be easy to implement).

> Dumitrescu, Adrian, and Joseph SB Mitchell. "Approximation algorithms for TSP with neighborhoods in the plane." *Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms (SODA)*. SIAM, 2001. ([PDF download link](http://www.ams.sunysb.edu/~jsbm/papers/tspn-jalgorithms.pdf))

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