Computation offloading between the edge and the cloud is an effective way for deployed service to fully utilize the resources at both sides for its QoS improvement and overall cost reduction. Although the offloading problem has been intensively studied in the context of mobile computing, existing algorithms in most cases cannot be effectively migrated to the edge-cloud environment because their inter-partition communication costs are always deemed as symmetric, and their intra-partition communication costs are often ignored, which, though reasonable to the traditional case, are not valid to our settings anymore. In this article, we propose a new algorithmic approach to the offloading problem in the edge-cloud environment, where a heterogeneous model is advocated to incorporate the communication cost between co-resident tasks while considering the asymmetry of communication costs between non-coresident tasks. We prove the offloading problem with respect to this model is NP-hard, and thereby designing an efficient algorithm to obtain a sub-optimal solution. Additionally, we also show that in a homogeneous case when the intra-partition and inter-partition communication costs between any pair of interactive tasks are symmetric, an optimal offloading algorithm can be devised by transforming the problem into a classical min-cut problem. We implemented and evaluated the algorithms by offloading a PageRank-based application in a controlled edge-cloud setting. Our empirical results show that the proposed algorithm for the heterogeneous case is always efficient to find a better offloading scheme, compared with the selected existing algorithms, while for the homogeneous case, the proposed solution can efficiently achieve the optimal strategy.