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Scheduling is adopted in various domains to assign jobs to resources, such that an objective is optimized. While schedules enable the analysis of the underlying system, publishing them also incurs a privacy risk. Recently, privacy attacks on schedules have been proposed, which may reveal sensitive information on the jobs by solving an inverse scheduling problem. In this work, we study the protection against such attacks. We formulate the problem of privacy-and-utility preservation of schedules, which bounds both, the privacy leakage and the loss in the utility of the schedule due to obfuscation. We address the problem based on a set of perturbation functions for schedules, study their instantiations for standard scheduling problems, and implement privacy-and-utility-aware publishing of a schedule using constraint programming. Experiments with synthetic and real-world schedules demonstrate the feasibility, robustness, and effectiveness of our mechanism.