Résumé
The Euclidean Traveling Salesman Problem with Neighborhoods (ETSPN) is a well-studied problem in computational geometry and has a wealth of results. In this problem, given a set of geometric neighborhoods (or regions), the goal is to compute a shortest route that visits at least one point of each neighborhood. The problem is a generalization of the standard Euclidean TSP and hence is also NP-hard, even when the neighborhoods are disjoint unit disks or unit squares in the plane. A longstanding open problem on this topic is the existence of PTASes for ETSPN with unit disk (and unit square) neighborhoods. Prior to this work, the best-known approximation factor for unit disks is 6.75, and PTASes are only known for the special case where the unit disks/squares are of bounded depth, i.e., each point lies in at most a constant number of disks.
In this paper, we resolve this open problem by giving the first PTASes for ETSPN with unit disks and unit squares. Our PTASes rely on new insights to the problem together with various classical tools such as Arora's technique and Baker's shifting technique.