In optical communication networks, appropriate path planning can improve the utilization of communication resources and bring a smooth communication experience to users. The following figure shows an inter-satellite optical communication network. User messages are sent from one terrestrial base station (nodes (4) to (7)) and transmitted through satellites (nodes (0) to (3)) in space to another terrestrial base station (nodes (4) to (7)). In the preceding figures, there are communication connections ( edges for short) between base stations and satellites and between satellites. The base stations and satellites are referred to as nodes . User messages are transmitted on these edges and referred to as flows . Some users may make video calls with friends, and some users may send short messages to their family members. Therefore, the message traffic (called flow rate ) of each user differs. There are many parallel edges (for example, edges (0), (1), and (2)) between two nodes, and the capacity of each edge also differs. Larger capacity indicates that more user messages can be transmitted, as well as shorter transmission distance indicates lower latency and better communication quality. Nodes also have their internal structure. As shown below, some edges inside a node cannot communicate with each other because these edges (constrained edge pair) are not fully connected. For example, edge (5) and edge (7) inside node (2) cannot communicate with each other, and therefore flows cannot pass through node (2) by traversing the two unconnected edges. Now, in the input network, the source node, target node, and required flow rate for each user flow are specified. Because network resources are limited, paths may not be successfully calculated for all user flows. We hope that you can provide a solution with the highest score. Note, that all edges are undirected, so flows may come in both directions, edges has both capaci