In the context of the Internet of Things (IoT), resource-constrained mobile edge computing (MEC) can no longer fully meet the needs of the rapidly growing number of mobile users; hence, cloud-edge collaborative computing has been developed. This paper focuses on the total energy consumption of the system and heterogeneity of scenarios, and a collaborative cloud-edge computation offloading approach with near real-time decision making is proposed. First, a general cloud-edge collaborative computation offloading model is abstracted from typical applications, and the energy consumption for edge and cloud offloading is calculated separately by considering both transmission and computational energy consumption. The problem is formulated as an integer linear program (ILP) with multidimensional resource constraints and is proven to be NP-hard. Then, a novel primal-dual computation offloading (PDCO) algorithm is designed to make near real-time offloading decisions one by one based on the sequential arrival of task requests. The approximation ratio of PDCO is derived through the weak duality property and the price-resource increment relationship. The experimental results show that under the guidance of total cost influenced by marginal prices, PDCO not only avoids blindly making offloading decisions but also effectively alleviates the shortage of resources on edge servers (ESs), approaching the optimal performance in terms of total energy consumption and resource utilization.

The intensity of radio waves decays rapidly with increasing propagation distance, and an edge server’s antenna needs more power to form a larger signal coverage area. Therefore, the power of the edge server should be controlled to reduce energy consumption. In addition, edge servers with capacitated resources provide services for only a limited number of users to ensure the quality of service (QoS). We set the signal transmission power for the antenna of each edge server and formed a signal disk, ensuring that all users were covered by the edge server signal and minimizing the total power of the system. This scenario is a typical geometric set covering problem, and even simple cases without capacity limits are NP-hard problems. In this paper, we propose a primal–dual-based algorithm and obtain an m-approximation result. We compare our algorithm with two other algorithms through simulation experiments. The results show that our algorithm obtains a result close to the optimal value in polynomial time.

In smart cities, the computing power and battery life of terminal devices (TDs) can be effectively enhanced by offloading tasks to nearby base stations (BSs) with richer resources. With the goal of TDs being fully served and achieving low-carbon energy savings for the system, this paper investigates task offloading in cloud-edge collaborative heterogeneous scenarios with multiple BSs and TDs. According to the proportional relationship between the energy and coverage radii of BSs, a complete coverage task offloading model with adjustable BS radii is proposed. The task offloading problem is formulated as an integer linear program with multidimensional resource constraints to minimize the sum of energy consumption of BS coverage, offloading tasks to BSs and the cloud data center (CC). Since this task offloading problem is NP-hard, two approximate algorithms with polynomial time complexity are designed based on the greedy strategy of seeking the most energy-effective disk and the primal–dual method of constructing primal feasible solutions according to dual feasible solutions. Experimental results show that both the greedy and primal–dual algorithms can achieve good approximation performance, but each of them has its own advantages due to different design principles. The former is superior in execution time and energy consumption, while the latter has advantages in balancing loads among BSs and alleviating core network bandwidth pressure.