Drop Cost and Wavelength Optimal Two-Period Grooming with Ratio 4

We study grooming for two-period optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the two-period grooming problem, during the first period of time there is all-to-all uniform traffic among n nodes, each request using 1/C of the bandwidth; and during the second period, there is all-to-all uniform traffic only among a subset V of v nodes, each request now being allowed to use 1/C′ of the bandwidth, where C′ < C. We determine the minimum drop cost (minimum number of ADMs) for any n, v and C = 4 and C′ ∈ {1, 2, 3}. To do this, we use tools of graph decompositions. Indeed the two-period grooming problem corresponds to minimizing the total number of vertices in a partition of the edges of the complete graph Kn into subgraphs, where each subgraph has at most C edges and where furthermore it contains at most C′ edges of the complete graph on v specified vertices. Subject to the condition that the two-period grooming has the least drop cost, the minimum number of wavelengths required is also determined in each case.