Upper bounds for the sensor subset selection problem

In this paper, we study the sensor subset selection problem with the determinant of the (Bayesian) Fisher information matrix (FIM) as the metric of estimation accuracy. As a combinatorial optimization problem, we analyze two well-known upper bounds for this problem: (i) the Lagrangian bound and (ii) the continuous bound. We show that the determinant of the FIM is a supermodular function from which it follows that the Lagrangian bound can be computed in polynomial time. We note that the continuous relaxation of the sensor subset selection problem can be transformed to a convex optimization problem from which it follows that the continuous bound is also computable in polynomial time. We also point to the benefit of using the natural selection process to solve the continuous relaxation of a variation of the sensor subset selection problem where sensors are allowed to make more than one measurement.