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Liveness is usually enforced by designing a supervisor that is supervisory in nature disabling events which otherwise would lead to the violation of the liveness specification. The supervisor is theoretically and practically expected to be maximally permissive such that it restricts the behavior of the plant (system under control) in a least restrictive manner while the liveness specification is not violated. However, the existence of a supervisory policy that enforces liveness in an arbitrary Petri net is undecidable. Based on elementary siphons of Petri nets, we develop a polynomial complexity approach to decide the existence of a maximally permissive monitor based liveness-enforcing Petri net supervisor for a subclass of Petri nets, S3PR that can well model a large class of flexible manufacturing systems. The results obtained in this paper are based on the computation of the set of elementary siphons and siphon composition operations in an S3PR in our previous work, which has been shown to be of polynomial complexity with respect to the size of an S3PR.