In this paper we present a exact linear formulation (mixed integer linear program) for designing a logical topology with no wavelength changers. Since the objective of our linear formulation is minimizing the maximum offered load on any link, which is called congestion, the resulting logical topology reflects the traffic intensities between the nodes. Our linear formulation yields a logical topology and routing and wavelength assignment for the logical links. Previously (see IEEE Journal on Selected Areas in Communications, vol.40, no.1, p.840-51, 1996) the problem of logical topology design is considered but the number of wavelengths the fiber can support is not a constraint. In this paper we take into consideration the number of wavelengths the fiber supports, the hop length of a logical link, and show the trade-offs involved in minimizing congestion. Since the whole problem is linearizable the solution obtained by relaxation of the integer constraints yields a lower bound on the congestion. This can be used to compare the efficiency of heuristic algorithms. We solve the problem exactly for a small size network and for large size networks we develop some heuristic algorithms to obtain a feasible solution using the solution obtained by relaxing the integer constraints. We introduce a cutting plane which enables us to reduce the previously obtained upper bounds on congestion. Numerical results for a six node wide area network and the National Science Foundation Network (NSFNET) are presented for various cases.
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