Bandwidth, Vertex Separators, and Eigenvalue Optimization
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A fundamental problem in numerical linear algebra consists in rearranging the rows and columns of a matrix in such a way that either the nonzero entries appear within a band of small width along the main diagonal, or such that the matrix has some block structure which is joined by only a few rows and columns. Such problems can be approached using graph partition techniques. From a practical point of view it is important that also large-scale instances can be dealt with. This rules out a direct application of the strong machinery for graph partition given by semidefinite optimization. We propose to use the weaker relaxations based on the Hoffman-Wielandt theorem, which lead to closed form bounds in terms of the Laplacian eigenvalues. We then try to improve these eigenvalue bounds by weight redistribution. This leads to nicely structured eigenvalue optimization problems. A similar approach has been used by Boyd, Diaconis and Xiao to increase the mixing rate of Markov chains. We use it to improve bounds on the bandwidth and the size of vertex separators in graphs. Moreover, the bounds can also be used to heuristically find good reorderings.
Key wordsVertex separator Bandwidth Semidefinite programming Eigenvalue optimization
Subject Classifications05C85 05C78
We thank an anonymous referee for several constructive suggestions to improve the presentation.
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