A Class of Weighted TSPs with Applications

Motivated by applications to poaching and burglary prevention, we define a class of weighted Traveling Salesman Problems on metric spaces. The goal is to output an infinite (though typically periodic) tour that visits the n points repeatedly, such that no point goes unvisited for "too long." More specifically, we consider two objective functions for each point x. The maximum objective is simply the maximum duration of any absence from x, while the quadratic objective is the normalized sum of squares of absence lengths from x. For periodic tours, the quadratic objective captures the expected duration of absence from x at a uniformly random point in time during the tour. The overall objective is then the weighted maximum of the individual points' objectives. When a point has weight w_x, the absences under an optimal tour should be roughly a 1/w_x fraction of the absences from points of weight 1. Thus, the objective naturally encourages visiting high-weight points more frequently, and at roughly evenly spaced intervals. We give a polynomial-time combinatorial algorithm whose output is simultaneously an O(log n) approximation under both objectives. We prove that up to constant factors, approximation guarantees for the quadratic objective directly imply the same guarantees for a natural security patrol game defined in recent work.