We study the autarky problem: given an energy forecast, a battery, and a set of energy-consuming jobs with time windows, decide whether all jobs can be scheduled without requiring external energy. We analyze the problem through the lens of job flexibility, defined as the number of time steps at which a job may be scheduled. We show that the problem is NP-hard already for flexibility two, even in restricted settings. On the positive side, we identify settings in which the problem is polynomial-time solvable, even for large flexibilities. Moreover, we obtain fixed-parameter tractability for combined parameters involving flexibility, such as the number of jobs. In contrast, we establish W-hardness when parameterized by maximum flexibility alone, even in a restricted setting. To complement our theoretical results, we formulate an integer linear program (ILP) that computes the minimum required external energy and evaluate it experimentally on instances derived from real-world energy-consumption and radiation data. The experiments indicate that increased job flexibility substantially reduces the need for external energy at moderate computational cost.

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