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Fog networks facilitate ultra-low latency through the use of data availability near the network edge in fog servers. Existing work in fog networks considers the objective of energy efficiency and low latency for internet-of-things (IoT) for resource allocation. These works provide solutions to energy efficiency and low latency resource allocation problem without consideration of secure communication. This article investigates the benefits of fog architecture from the perspective of three promising technologies namely device-to-device (D2D) communication, caching, and physical layer security. We propose security provisioning followed by mode selection for D2D-assisted fog networks. The secrecy rate maximization problem is formulated first, which belongs to mixed-integer nonlinear programming (MINLP) problem. It is NP-hard, that is why an exhaustive search for finding the solution is complex. Keeping in view the complexity, a nonlinear technique namely outer approximation algorithm (OAA) is applied. OAA is a traditional algorithm, whose results are compared with the proposed heuristic algorithm, namely the security heuristic algorithm (SHA). Performance of the network is observed for the different numbers of eavesdroppers, IoT nodes, and fog nodes. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompatible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is Algorithm 2 which is 2-approximate for the problem Qmjpj = 1, G = bisubquarticjCmax. The second one is Algorithm 3 which is 4-approximate for the problem Qmjpj = 1, G = bisubquarticjSCj, where m 2 {2, 3, 4}. The theory behind the proposed algorithms is based on the properties of 2-coloring with maximal coloring width, and on the properties of ideal machine, an abstract machine that we introduce in this paper. [ABSTRACT FROM AUTHOR]