The Complexity of Tracking a Stopping Time

We present a generalization of the well-known Bayesian change-point detection problem. Specifically, let {(X_i,Y_i)}_iges1 be a sequence of pairs of random variables, and let S be a stopping time with respect to {X_i}_iges1. We assume that the (X_i, Y_i)´s take values in the same finite alphabet X times Y. For a fixed kappa ges 1, we consider the problem of finding a stopping time T < kappa with respect to {Y_i}_iges1 that optimally tracks S, in the sense that T minimizes the average reaction time E(T - S)⁺ , while it keeps the false-alarm probability P(T < S) below a given threshold alpha isin [0,1]. In previous work, we presented an algorithm that computes the optimal expected reaction times for all alpha isin [0,1] such that alpha ges P(S > kappa), and constructs the associated optimal stopping times T. In this paper, we provide a sufficient condition on {(X_i,Y_i)}_iges1 and S under which the algorithm running time is polynomial in kappa, and we illustrate this condition with two examples: a Bayesian change-point problem and a pure tracking stopping time problem.