We call the gaps cuts and the maximum number of wires that can be placed between a cut the capacity of the cut. We define the actual number of wires across a cut to be the flow of the cut. If we can assign proper flows to all cuts so that all flows are less than or equal to its respective capacities, we have found a solution. The flows are not arbitrary because wires cannot be created or destroyed at arbitrary points. We call the small pin set in a pin group the source set and the other set the sink set.
Wires can only originate at source pins and terminate at sink pins.
In graph theory, this is called a network flow problem. The flow router builds a routing network for a design. The routing network consists of a set of vertices and a set of edges. The vertices represent the regions enclosed by cuts. Two vertices are linked by an edge if their corresponding regions share a cut.