Recognizing sparse perfect elimination bipartite graphs

Abstract

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into nonzeros to preserve sparsity. Perfect elimination bipartite graphs are closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a nonzero. Existing literature on the recognition of these graphs mainly focuses on time complexity. For n×n matrices with m nonzero elements, the best known algorithm runs in time . However, the space complexity also deserves attention: it may not be worthwhile to look for suitable pivots for a sparse matrix if this requires space. We present two new recognition algorithms for sparse instances: one with a time complexity in space and one with a time complexity in space. Furthermore, if we allow only pivots on the diagonal, our second algorithm is easily adapted to run in time . © 2011 Springer-Verlag Berlin Heidelberg.