Average minimum distance to visit a subset of random points in a compact region

This paper seeks an analytical estimate of the expected distance for visiting an arbitrary subset of independently and uniformly distributed random points within a compact region. This problem has many real-world application contexts such as the emerging on-demand transportation and logistics services (e.g., ridesharing, customized buses). The lower bounds of the expected optimal tour length are analytically derived by considering a so-called "trapping effect", which explicitly addresses probabilistically the situation that some of the tour legs must connect points that are not neighbors. A parametric approach is developed to estimate the expected optimal tour length for both Euclidean and rectilinear metrics. Numerical experiments demonstrate the validity of these bounds, as well as the closeness of the proposed estimator to simulated results.