On the submodularity of sensor scheduling for estimation of linear dynamical systems

In this paper, we study the problem of using an energy constrained sensor network to estimate the state of a linear dynamical system. The state estimate is computed using a Kalman filter and the goal is to choose a subset of sensors at each time step so as to minimize the a posteriori error covariance. Recent work has indicated that the simple greedy algorithm, which chooses the sensor at each time step that maximizes the error covariance reduction, outperforms many other known scheduling algorithms. In addition, it has been suggested that the cost function mapping a sensor sequence to an error covariance cost is submodular; this would imply that the greedy algorithm provides a near optimal sensor schedule. As a negative result, we show that the sensor schedule cost is not, in general, a submodular function. This contradicts an established result. We argue that given a linear dynamical system, it is computationally intractable to determine if it will yield a submodular cost. Thus, we provide sufficient and easily checkable conditions under which the dynamical system yields a submodular cost, and thus performance guarantees for the greedy schedule.