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This paper studies a mobile edge computing (MEC) system in which two mobile devices are energized by the wireless power transfer (WPT) from an access point (AP) and they can offload part or all of their computation-intensive latency-critical tasks to the AP connected with an MEC server or an edge cloud. This harvest-then-offload protocol operates in an optimized time-division manner. To overcome the doubly near-far effect for the farther mobile device, cooperative communications in the form of relaying via the nearer mobile device is considered for offloading. Our aim is to minimize the AP's total transmit energy subject to the constraints of the computational tasks. We illustrate that the optimization is equivalent to a min-max problem, which can be optimally solved by a two-phase method. The first phase obtains the optimal offloading decisions by solving a sum-energy-saving maximization problem for given an …