  We examine a control problem where the states of the components of a system
deteriorate after a disruption, if they are not being repaired by an entity.
There exist a set of dependencies in the form of precedence constraints between
the components, captured by a directed acyclic graph (DAG). The objective of
the entity is to maximize the number of components whose states are brought
back to the fully repaired state within a given time. We prove that the general
problem is NP-hard, and therefore we characterize near-optimal control policies
for special instances of the problem. We show that when the deterioration rates
are larger than or equal to the repair rates and the precedence constraints are
given by a DAG, it is optimal to continue repairing a component until its state
reaches the fully recovered state before switching to repair any other
component. Under the aforementioned assumptions and when the deterioration and
the repair rates are homogeneous across all the components, we prove that the
control policy that targets the healthiest component at each time-step while
respecting the precedence and time constraints fully repairs at least half the
number of components that would be fully repaired by an optimal policy.
Finally, we prove that when the repair rates are sufficiently larger than the
deterioration rates, the precedence constraints are given by a set of disjoint
trees that each contain at most k nodes, and there is no time constraint, the
policy that targets the component with the least value of health minus the
deterioration rate at each time-step while respecting the precedence
constraints fully repairs at least 1/k times the number of components that
would be fully repaired by an optimal policy.
