Current network applications, such as utility networks (gas, water, electricity, and 4G/5G), the Internet of Things (IoT), social networks, and supply chains, are all based on binary state networks. Reliability is one of the most commonly used tools for evaluating network performance, and the minimal path (MP) is a basic algorithm for calculating reliability. However, almost all existing algorithms assume that all undirected arcs are homogeneous; that is, the probability of an arc from nodes a to b is equal to that from nodes b to a. Therefore, based on MPs, the binary-addition-tree algorithm (BAT), and the inclusion-exclusion technique (IET), a novel recursive inclusion-exclusion technology algorithm known as recursive BAT-based IET (RIE) is proposed to solve the heterogeneous-arc binary-state network reliability problem to overcome the above obstacles in applications. The computational complexity of the proposed RIE is analyzed using an illustrative example. Finally, 11 benchmark problems are used to verify the performance of RIE.