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The Euclidean TSP with neighborhoods (TSPN) is the following problem: Given aset R of k regions, find a shortest tour that visits at least one point fromeach region. We study the special cases of disjoint, connected, alpha-fatregions (i.e., every region P contains a disk of diameter diam(P)/alpha) anddisjoint unit disks.  For the latter, Dumitrescu and Mitchell proposed an algorithm based onMitchell's guillotine subdivision approach for the Euclidean TSP and claimed itto be a PTAS. However, their proof contains a severe gap, which we will closein the following. Bodlaender et al. remark that their techniques for theminimum corridor connection problem carry over to the TSPN and yield analternative PTAS for this problem.  For disjoint connected alpha-fat regions of varying size, Mitchell proposed aslightly different PTAS candidate. We will expose several further problems andgaps in this approach. Some of them we can close, but overall, for alpha-fatregions, the existence of a PTAS for the TSPN remains open.

The guillotine approach for TSP with neighborhoods revisited