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Edge-cloud computing is an efficient approach to address the high latency issue in mobile cloud computing for service provisioning, by placing several computing resources close to end devices. To improve the user satisfaction and the resource efficiency, this paper focuses on the task offloading and service caching problem for providing services by edge-cloud computing. This paper formulates the problem as a constrained discrete optimization problem, and proposes a hybrid heuristic method based on Particle Swarm Optimization (PSO) to solve the problem in polynomial time. The proposed method, LMPSO, exploit PSO to solve the service caching problem. To avoid PSO trapping into local optimization, LMPSO adds lévy flight movement for particle updating to improve the diversity of particle. Given the service caching solution, LMPSO uses a heuristic method with three stages for task offloading, where the first stage tries to make full use of cloud resources, the second stage uses edge resources for satisfying requirements of latency-sensitive tasks, and the last stage improves the overall performance of task executions by re-offloaded some tasks from the cloud to edges. Simulated experiment results show that LMPSO has upto 156% better user satisfaction, upto 57.9% higher resource efficiency, and upto 155% greater processing efficiency, in overall, compared with other seven heuristic and meta-heuristic methods.