We give new approximation algorithms for robot routing problems that are variants of the classical traveling salesperson problem (TSP). We are to find a path for a robot, moving at speed at most s, to visit a set V={v1,…,vn} of sites, each having an associated time window of availability, [ri,di], between a release time ri and a deadline di.
In the time-window prize collecting problem (TWPC), the objective is to maximize the number of sites visited within their time windows.
In the time-window TSP problem (TWTSP), the objective is to minimize the length of a path that visits all of the sites V within their respective time windows, if it is possible to do so within the speed bound s.
For sites on a line, we give approximation algorithms for TWPC and TWTSP that produce paths that visit sites vi at times within the relaxed time windows [ri−εLi,di+εLi], for fixed ε>0, where Li=di−ri; the running time is O((nLmax)O(logLmaxlog(1+ε))), where Lmax=maxiLi.
For TWPC, the computed path visits at least k∗ (the cardinality of an optimal solution to TWPC) sites; for TWTSP, the computed path is of length at most λ∗ (the length of an optimal TWTSP solution).
For general instances of sites in a metric space, we give approximation algorithms that apply to instances with certain special structure of the time windows (that they are ``dyadic'' or that they are ``elementary''), giving paths whose lengths are within a bounded factor of the optimal length, λ∗(s), for the given speed s, while relaxing the speed to be a factor greater than s; for arbitrary time windows, we give an O(logn)-approximation for TWTSP, assuming unbounded speed (s=∞).