Abstract
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.
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References
Anstreicher, K.M., Wolkowicz, H.: On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. 22, 41–55 (2000)
Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing Markov chain on a graph. SIAM Rev. 46(4), 667–689 (2004)
de Klerk, E., Nagy, M., Sotirov, R.: On semidefinite programming bounds for graph bandwidth. Technical report, Tilburg University, Netherlands (2011)
de Sousa, C., Balas, E.: The vertex separator problem: algorithms and computations. Math. Program. (A) 103, 609–631 (2005)
Göhring, F., Helmberg, C., Reiss, S.: Graph realizations associated with minimizing the maximum eigenvalue of the Laplacian. Math. Program. (A) 131, 95–111 (2012)
Helmberg, C., Mohar, B., Poljak, S., Rendl, F.: A spectral approach to bandwidth and separator problems in graphs. Linear Multilinear Algebra 39, 73–90 (1995)
Hoffman, A.J., Wielandt, H.W.: The variation of the spectrum of a normal matrix. Duke Math. J. 20, 37–39 (1953)
Juvan, M., Mohar, B.: Optimal linear labelings and eigenvalues of graphs. Discret. Appl. Math. 36, 153–168 (1992)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)
Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18(1), 223–241 (2007)
Povh, J., Rendl, F.: Approximating non-convex quadratic programs by semidefinite and copositive programming. In: Neralic, L., Boljuncic, V., Soric, K. (eds.) Proceedings of the 11th International Conference on Operational Research, Pula, pp. 35–45. Croation Operations Research Society (2008)
Rendl, F.: Semidefinite relaxations for integer programming. In: Jünger, M., Liebling, Th.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 687–726. Springer, Berlin/Heidelberg (2009)
Rendl, F., Wolkowicz, H.: A projection technique for partitioning the nodes of a graph. Ann. Oper. Res. 58, 155–179 (1995)
von Neumann, J.: Some matrix inequalities and metrization of matrix space, pp. 205–219. Reprinted in: John von Neumann: Collected Works, vol. 4. MacMillan (1962/1937)
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We thank an anonymous referee for several constructive suggestions to improve the presentation.
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Rendl, F., Lisser, A., Piacentini, M. (2013). Bandwidth, Vertex Separators, and Eigenvalue Optimization. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_14
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DOI: https://doi.org/10.1007/978-3-319-00200-2_14
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