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When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For n × n matrices with m non-zero elements, the currently best known algorithm has a time complexity of O n 3 / log n. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires Ω n 2 space. We present two new algorithms for the recognition of sparse instances: one with a O (nm) time complexity in Θ n 2 space and one with a O m 2 time complexity in Θ (m) space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time O (nm).

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When solving a system of equations, it can be beneficial not to solve it in its entirety at once, but rather to decompose it into smaller subsystems that can be solved in order. Based on a bisimplicial graph representation we analyze the parameterized complexity of two problems central to such a decomposition: The Free Square Block problem related to finding smallest subsystems that can be solved separately, and the Bounded Block Decomposition problem related to determining a decomposition where the largest subsystem is as small as possible. We show both problems to be W [1]-hard. Finally we relate these problems to crown structures and settle two open questions regarding them using our results.