The emerging software-defined networks (SDNs) have made centralized flow control practical. In this paper, we study the flow control problem with consideration of path constraints and user flow requirements. Two problems are formulated according to path constraints, namely, the flow control with independent paths (FC-IDP) problem and the flow control with destination-rooted trees (FC-DRT) problem. It is shown that a flow rate allocation vector is weighted max-min fair if and only if each flow has a weighted bottleneck link. The optimality conditions are analyzed for the FC-IDP problem without fairness constraints. A weighted water filling algorithm is proposed for flow control with fairness constraints and predetermined flow paths. Without fairness constraints, evolutionary strategy is proposed to optimize the flow paths and flow rate allocation vectors. The algorithms are tested on random networks and it is shown that fairness constraints may have a great impact on flow control performance.