  This paper studies the robustness of observability of a linear time-invariant
system under sensor failures from a computational perspective. To be precise,
the problem of determining the minimum number of sensors whose removal can
destroy system observability, as well as the problem of determining the minimum
number of state variables that need to be prevented from being directly
measured by the existing sensors to destroy observability, is investigated. The
first one is closely related to the ability of unique state reconstruction of a
system under adversarial sensor attacks, and the dual of both problems are in
the opposite direction of the well-studied minimal controllability problems. It
is proven that all these problems are NP-hard, both for a numerical system and
a structured system, even restricted to some special cases. It is also shown
that the first problems both for a numerical system and a structured one share
a cardinality-constrained submodular minimization structure, for which there is
no known constant or logarithmic factor approximation in general. On the other
hand, for the first two problems, under a reasonable assumption often met by
practical systems, that the eigenvalue geometric multiplicities of the
numerical systems or the matching deficiencies of the structured systems are
bounded by a constant, by levering the rank-one update property of the involved
rank function, it is possible to obtain the corresponding optimal solutions by
traversing a subset of the feasible solutions. We show such a method has
polynomial time complexity in the system dimensions and the number of sensors.
