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Data compression (DC) has the potential to significantly improve the computation offloading performance in hierarchical fog-cloud systems. However, it remains unknown how to optimally determine the compression ratio jointly with the computation offloading decisions and the resource allocation. This optimization problem is studied in this paper where we aim to minimize the maximum weighted energy and service delay cost (WEDC) of all users. First, we consider a scenario where DC is performed only at the mobile users. We prove that the optimal offloading decisions have a threshold structure. Moreover, a novel three-step approach employing convexification techniques is developed to optimize the compression ratios and the resource allocation. Then, we address the more general design where DC is performed at both the mobile users and the fog server. We propose three algorithms to overcome the strong …