We develop a joint offloading and resource allocation framework for a multi-layer cooperative fog computing network, aiming to minimize the total energy consumption of multiple mobile devices subject to their service delay requirements. The resulting optimization involves both binary (offloading decisions) and real variables (resource allocations), making it an NP-hard and computationally intractable problem. To tackle it, we first propose an improved branch-and-bound algorithm (IBBA) that is implemented in a centralized manner. However, due to the large size of the cooperative fog computing network, the computational complexity of the proposed IBBA is relatively high. To speed up the optimal solution searching as well as to enable its distributed implementation, we then leverage the unique structure of the underlying problem and the parallel processing at fog nodes. To that end, we propose a distributed framework, namely feasibility finding Benders decomposition (FFBD), that decomposes the original problem into a master problem for the offloading decision and subproblems for resource allocation. The master problem (MP) is then equipped with powerful cutting-planes to exploit the fact of resource limitation at fog nodes. The subproblems (SP) for resource allocation can find their closed-form solutions using our fast solution detection method. These (simpler) subproblems can then be solved in parallel at fog nodes. The numerical results show that the FFBD always returns the optimal solution of the problem with significantly less computation time (e.g., compared with the centralized IBBA approach). The FFBD with the fast solution detection method, namely FFBD-F, can reduce up to $60\%$ and $90\%$ of computation time, respectively, compared with those of the conventional FFBD, namely FFBD-S, and IBBA.

中文翻译：

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具有延迟约束的多层协同雾计算网络的最优能效

我们为多层协作雾计算网络开发了一个联合卸载和资源分配框架，旨在最大限度地减少受服务延迟要求影响的多个移动设备的总能耗。由此产生的优化涉及二进制（卸载决策）和实际变量（资源分配），使其成为一个 NP 难和计算上难以处理的问题。为了解决这个问题，我们首先提出了一种以集中方式实现的改进的分支定界算法（IBBA）。然而，由于协同雾计算网络的规模较大，所提出的IBBA的计算复杂度相对较高。为了加速最优解的搜索并使其分布式实现，然后我们利用底层问题的独特结构和雾节点的并行处理。为此，我们提出了一个分布式框架，即可行性发现 Benders 分解（FFBD），它将原始问题分解为卸载决策的主问题和资源分配的子问题。然后，主问题 (MP) 配备了强大的切割平面，以利用雾节点资源限制的事实。用于资源分配的子问题 (SP) 可以使用我们的快速解决方案检测方法找到其封闭形式的解决方案。然后可以在雾节点并行解决这些（更简单的）子问题。数值结果表明，FFBD 总是以明显更少的计算时间返回问题的最优解（例如，与集中式 IBBA 方法相比）。