  This article studies two problems related to observability and efficient
constrained sensor placement in linear time-invariant discrete-time systems
with partial state observations. (i) We impose the condition that both the set
of outputs and the state that each output can measure are pre-specified. We
establish that for any fixed \(k > 2\), the problem of placing the minimum
number of sensors/outputs required to ensure that the structural observability
index is at most \(k\), is NP-complete. Conversely, we identify a subclass of
systems whose structures are directed trees with self-loops at every state
vertex, for which the problem can be solved in linear time. (ii) Assuming that
the set of states that each given output can measure is given, we prove that
the problem of selecting a pre-assigned number of sensors in order to maximize
the number of states of the system that are structurally observable is also
NP-hard. As an application, we identify suitable conditions on the system
structure under which there exists an efficient greedy strategy, which we
provide, to obtain a \((1-\frac{1}{e})\)-approximate solution. An illustration
of the techniques developed for this problem is given on the benchmark IEEE
118-bus power network containing roughly \(400\) states in its linearized
model.
