# TSP With Point Locational Uncertainty: The Adversarial Model

## March 7, 2017

We study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space $(X, d)$ and a set of subsets $R = \{R_1, R_2, ... , R_n\} : R_i \subseteq X$ the goal is to devise an ordering of the regions, $\sigma_R$, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by $\sigma_R$ is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some known probability distribution, here, we focus on the adversarial model where once $\sigma_R$ is prescribed, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words we assume the worst point possible within each region, with respect to the ordering of the regions prescribed, will be chosen by an adversary as part of an offline problem. We give a $3 + \frac1n$-approximation when $R$ is a set arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are disjoint unit line segments of the same orientation, and a polynomial time approximation scheme (PTAS) for the important special case when $R$ is a set of disjoint unit disks in the plane.