The anchored rescheduling problem, recently introduced in the literature, is to find a schedule under precedence constraints with a maximum number of prescribed starting times. Namely, prescribed starting times may correspond to a former schedule that must be modified while maintaining a maximum number of starting times unchanged. In the present work two extensions are investigated. First we introduce a new tolerance feature, so that starting times can be considered as unchanged when modified less than a tolerance threshold. The sensitivity of the anchored rescheduling problem to tolerance is studied. Second we consider generalized precedence constraints, which include, e.g., deadline constraints. Altogether this leads to a more realistic rescheduling problem. The main result is to show that the problem is polynomial. We discuss how to benefit from the polynomiality result in a machine scheduling environment.