TSP With Locational Uncertainty: The Adversarial Model

In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the {em 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 probability distribution, here, we focus on the {em adversarial model} in which once the choice of $sigma_R$ is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions $R$ that is optimal with respect to the worst point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a $3$-approximation when $R$ is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which $R$ is a set of disjoint unit disks in the plane.