In implementations of item transfer control systems, a computing device implements a transfer system to receive input data describing types of requested items and corresponding quantities of the types of requested items to receive at each of a plurality of destination sites and types of available items and corresponding quantities of the types of available items that are available at each of a plurality of source sites. The transfer system constructs a flow network having a source node for each of the plurality of the source sites and a destination node for each of the plurality of the destination sites. An integral approximate solution is generated that transfers the corresponding quantities of the types of requested items to each of the plurality of the destination sites using a maximum flow solver and the flow network. The transfer system causes transferences of the corresponding quantities of the types of requested items to each of the plurality of the destination sites based on the integral approximate solution.