  This paper studies the problem of, given the structure of a linear-time
invariant system and a set of possible inputs, finding the smallest subset of
input vectors that ensures system's structural controllability. We refer to
this problem as the minimum constrained input selection (minCIS) problem, since
the selection has to be performed on an initial given set of possible inputs.
We prove that the minCIS problem is NP-hard, which addresses a recent open
question of whether there exist polynomial algorithms (in the size of the
system plant matrices) that solve the minCIS problem. To this end, we show that
the associated decision problem, to be referred to as the CIS, of determining
whether a subset (of a given collection of inputs) with a prescribed
cardinality exists that ensures structural controllability, is NP-complete.
Further, we explore in detail practically important subclasses of the minCIS
obtained by introducing more specific assumptions either on the system dynamics
or the input set instances for which systematic solution methods are provided
by constructing explicit reductions to well known computational problems. The
analytical findings are illustrated through examples in multi-agent
leader-follower type control problems.
