The guillotine approach for TSP with neighborhoods revisited


The Euclidean TSP with neighborhoods (TSPN) is the following problem: Given a set R of k regions (subsets of R), find a shortest tour that visits at least one point from each region. We study the special cases of disjoint, connected, α-fat regions (i.e., every region P contains a disk of diameter diam(P ) α ) and disjoint unit disks. For the latter, Dumitrescu and Mitchell [4] proposed an algorithm based on Mitchell’s guillotine subdivision approach for the Euclidean TSP [9], and claimed it to be a PTAS. However, their proof contains a severe gap, which we will close in the following. Bodlaender et al. [2] remark that their techniques for the minimum corridor connection problem based on Arora’s PTAS for TSP [1] carry over to the TSPN and yield an alternative PTAS for this problem. For disjoint connected α-fat regions of varying size, Mitchell [10] 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 α-fat regions, the existence of a PTAS for the TSPN remains open.

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@article{Spirkl2013TheGA, title={The guillotine approach for TSP with neighborhoods revisited}, author={Sophie Theresa Spirkl}, journal={CoRR}, year={2013}, volume={abs/1312.0378} }