The paper examines the problem of finding the optimal sequencing decisions that a repairing agency should follow to repair infrastructure components, subject to precedence constraints between them, after disasters. Specifically, it considers scenarios where the health of an infrastructure component continues to deteriorate unless it is being repaired, until a certain point where the component fails irreversibly. It captures precedence constraints between components (indicating that one component must be fully repaired before the other can start to be repaired) by a directed acyclic graph. It characterizes approximation algorithms for sequencing decisions depending on the relationship between the rates of repair and deterioration.