We study the complexity of the reachability problem for a  new subclass of Petri nets called simple-circuit Petri nets, which properly  contains several well known subclasses such as conflict-free, BPP, normal   Petri nets and more. A new decomposition approach is applied to  developing an integer linear programming formulation for characterizing  the reachability sets of such Petri nets. Consequently, the reachability  problem is shown to be NP-complete. The model checking problem for  some temporal logics is also investigated for simple-circuit Petri nets