Target-Value Search (& II)
What follows is taken from Tsuyoshi Ito's comment. 

In the case of DAG, all paths can be enumerated in polynomial space and therefore the problem cannot be EXPTIME-complete unless PSPACE=EXPTIME. The natural decision version (given a DAG G, its vertices s and t, and numbers x and y, decide whether there is a path from s to t whose length is between x and y) is in NP.