The Bandwidth Problem asks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions to bound the bandwidth of a graph. Our approach contains prior approaches for bounding the bandwidth as special cases. By varying sizes of partitions, we have a trade-off between quality of bounds and efficiency of computing them. To compute lower bounds, we derive several semidefinite programming relaxations. We evaluate the performance of our approach on a carefully selected set of benchmark instances.

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