The concept of the limiting cut, introduced in Section 6.3.1.2, stems from the Min Cut–Max Flow relation in multicommodity flow problems. We first give a brief summary of this problem and then present a heuristic for finding limiting cuts.

The Multicommodity Flow Problem and Limiting Cuts

In the most common version of the multicommodity flow problem, a set of demands are prescribed between source-destination node pairs in a network with a given topology and link capacities. (Each source-destination demand is known as a commodity, and the network can be anything—gas pipelines, airline routes, highways, and so on.) The basic issue is whether the prescribed demands can be satisfied within the capacity constraints; that is, whether all commodities can be routed through the network (in a bifurcated manner if necessary) so that the total flow of all commodities on each link does not exceed its capacity. If so, the demands are said to be feasible.

In wavelength-routed networks (WRNs), the commodities (demands) are LCs, each requiring one λ-channel, so the capacity of a cut Ci is FiW, where Fi is the number of fiber pairs in the cut and W is the number of available wavelengths. Because a channel in a WRN is a single point-to-point entity, bifurcated routing is not permitted in a WRN. (An exception would be a case in which several λ-channels are required to carry the flow on one LC.)

The relations between cut capacities and feasible demands were stated in Section A.1.8 for the single-commodity case. In the multicommodity case, which is of interest here, the relations are considerably more complex.