Abstract
In all-optical networks, several communications can be transmitted through the same fiber link provided that they use different wavelengths. The MINIMUM ALL-OPTICAL ROUTING problem (given a list of pairs of nodes standing for as many point to point communication requests, assign to each request a route along with a wavelength so as to minimize the overall number of assigned wavelengths) has been paid a lot of attention and is known to be NP-hard. Rings, trees and meshes have thus been investigated as specific networks, but leading to just as many NP-hard problems. This paper investigates row-column routings in meshes (paths are allowed one turn only). We first show the MINIMUM LOAD ROW-COLUMN ROUTING problem to be NP-hard but 2-APX (more generally, the MINIMUM LOAD k-CHOICES ROUTING problem is NP-hard but k-APX), then that the MINIMUM ROW-COLUMN PATHS COLOURING problem is 4-APX (more generally, any d-segmentable routing of load L in a hypermesh of dimension d can be coloured with 2d(L-1)+1 colours at most). From there, we prove the MINIMUM ALL-OPTICAL ROW-COLUMN ROUTING problem to be APX.