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In this paper, we propose an efficient approximation approach, called Lagrangean relaxation with heuristics (LRH), aimed to resolve routing and wavelength assignment (RWA) for multi-granularity WDM networks facilitating fiber, waveband, and lambda switching capabilities. The task is first formulated as a combinatorial optimization problem in which the bottleneck link utilization is to be minimized. The LRH approach performs constraint relaxation and derives a lower-bound solution index according to a set of Lagrangean multipliers generated through subgradient-based iterations. In parallel, using the generated Lagrangean multipliers, the LRH approach employs a new heuristic algorithm to arrive at a near-optimal upper-bound solution. With lower and upper bounds, we delineate the performance of LRH with respect to accuracy and convergence speed under different parameter settings. We further draw comparisons between LRH and a typical linear programming (LP) approach via experiments over the widely-used NSFNET and three randomly generated networks. Numerical results demonstrate that LRH outperforms the LP approach in both accuracy and computational time complexity particularly for larger sized networks.