# Recognizing sparse perfect elimination bipartite graphs

M.J. Bomhoff

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

1 Citation (Scopus)

### 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 $n \times n$ matrices with $m$ nonzero elements, the best known algorithm runs in time $O(n^3/\log n)$. However, the space complexity also deserves attention: it may not be worthwhile to look for suitable pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new recognition algorithms for sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm is easily adapted to run in time $O(n m)$
Original language Undefined Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011 A. Kulikov, N. Vereshchagin Heidelberg Springer 443-455 13 978-3-642-20711-2 https://doi.org/10.1007/978-3-642-20712-9_35 Published - Jun 2011

### Publication series

Name Lecture Notes in Computer Science Springer Verlag 6651 0302-9743 1611-3349

### Keywords

• METIS-284944
• Perfect elimination – algorithms – bipartite graphs – sparse graphs
• EWI-21125
• IR-79141

### Cite this

Bomhoff, M. J. (2011). Recognizing sparse perfect elimination bipartite graphs. In A. Kulikov, & N. Vereshchagin (Eds.), Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011 (pp. 443-455). (Lecture Notes in Computer Science; Vol. 6651). Heidelberg: Springer. https://doi.org/10.1007/978-3-642-20712-9_35
Bomhoff, M.J. / Recognizing sparse perfect elimination bipartite graphs. Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011. editor / A. Kulikov ; N. Vereshchagin. Heidelberg : Springer, 2011. pp. 443-455 (Lecture Notes in Computer Science).
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title = "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 $n \times n$ matrices with $m$ nonzero elements, the best known algorithm runs in time $O(n^3/\log n)$. However, the space complexity also deserves attention: it may not be worthwhile to look for suitable pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new recognition algorithms for sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm is easily adapted to run in time $O(n m)$",
keywords = "METIS-284944, Perfect elimination – algorithms – bipartite graphs – sparse graphs, EWI-21125, IR-79141",
author = "M.J. Bomhoff",
note = "10.1007/978-3-642-20712-9_35",
year = "2011",
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isbn = "978-3-642-20711-2",
series = "Lecture Notes in Computer Science",
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editor = "A. Kulikov and N. Vereshchagin",
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Bomhoff, MJ 2011, Recognizing sparse perfect elimination bipartite graphs. in A Kulikov & N Vereshchagin (eds), Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011. Lecture Notes in Computer Science, vol. 6651, Springer, Heidelberg, pp. 443-455. https://doi.org/10.1007/978-3-642-20712-9_35

Recognizing sparse perfect elimination bipartite graphs. / Bomhoff, M.J.

Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011. ed. / A. Kulikov; N. Vereshchagin. Heidelberg : Springer, 2011. p. 443-455 (Lecture Notes in Computer Science; Vol. 6651).

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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Bomhoff MJ. Recognizing sparse perfect elimination bipartite graphs. In Kulikov A, Vereshchagin N, editors, Computer Science – Theory and Applications: Proceedings 6th International Computer Science Symposium in Russia, CSR 2011. Heidelberg: Springer. 2011. p. 443-455. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-642-20712-9_35