Static Optimal Sensor Selection via Linear Integer Programming: The Orthogonal Case

We consider the static optimal sensor selection problem, where we optimally select d sensors among s possible sensors with d <; s. Under the assumption that the s sensors are mutually orthogonal to each other, we cast the optimal sensor selection problem as a linear integer program (LIP) that corresponds to minimization of the trace of the linear least-squares estimation error covariance. We show that even though general LIPs are NP-hard, our problem can be solved in polynomial time as a linear program; hence, it is not necessary to go for suboptimal solutions. This is due to the associated integral convex polyhedron constraint set followed by its total unimodularity property. We provide simulation results to demonstrate polynomial-time solvability of the corresponding problem with the orthogonality condition as well as additional sensor selection constraints.