An important issue in an edge computing (EC) network is to increase the utilities of the end users concurrently accessing the computation resources. In this paper, we consider the task offloading in EC-enabled networks where the end users efficiently utilize the dispersed computation and communication resources in a multi-path multi-hop manner. We propose a binary optimization framework that generalizes multi-hop wireless EC task offloading as jointly making decisions of server selecting and traffic routing in networks of arbitrary topology (JoSRAT). We further develop an approximation algorithm JoSRAT that enables for a fully distributed implementation together with the worst-case performance guarantees. Interestingly, our proposed distributed algorithm achieves nearly optimal in the numerical evaluations, significantly outperforming the worst-case guarantees. The proposed algorithm also outperforms a widely-used heuristic, i.e., First Fit, in terms of computational time complexity, indicating the superior capability of the proposed framework.