The guillotine approach for TSP with neighborhoods revisited

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