Abstract
The vertex ranking problem is closely related to the problem of finding the elimination tree of minimum height for a given graph. This implies that the problem has applications in the parallel Cholesky factorization of matrices. We describe the connection between this model of graph coloring and the matrix factorization. We also present a polynomial time algorithm for finding edge ranking of complete bipartite graphs. We use it to design an O(m 2 + d) algorithm for edge ranking of graphs obtained by removing O(log m) edges from a complete bipartite graph, where d is a fixed number. Then we extend our results to complete k-partite graphs for any fixed k>2. In this way we give a new class of matrix factorization instances that can be optimally solved in polynomial time.
Supported in part by KBN grant 4T11C 04725
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References
Bodlaender, H., Deogun, J.S., Jansen, K., Kloks, T., Kratsch, D., Müller, H., Tuza, Z.: Rankings of graphs. SIAM J.Discrete Math. 11, 168–181 (1998)
Deogun, J.S., Kloks, T., Kratsch, D., Müller, H.: On the vertex ranking problem for trapezoid, circular-arc and other graphs. Discrete Appl. Math. 98, 39–63 (1999)
Iyer, A.V., Ratliff, H.D., Vijayan, G.: Parallel assembly of modular products - an analysis, Tech. Report 88-06, Georgia Institute of Technology (1988)
Katchalski, M., McCaugh, W., Seager, S.: Ordered colourings. Discrete Math. 142, 141–154 (1995)
Lam, T.W., Yue, F.L.: Edge ranking of graphs is hard. Discrete Appl. Math. 85, 71–86 (1998)
Lam, T.W., Yue, F.L.: Optimal edge ranking of trees in linear time. In: Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 436–445 (1998)
Leiserson, C.E.: Area-efficient graph layout (for VLSI). In: Proc. 21st Ann. IEEE Symp. on Foundations of Computer Science, pp. 270–281 (1980)
Liu, J.W.H.: The role of elimination trees in sparse factorization. SIAM J.Matrix Analysis and Appl. 11, 134–172 (1990)
Manne, F.: Reducing the height of an elimination tree through local reorderings. Tech. Report 51, University of Bergen, Norway (1991)
Pothen, A.: The complexity of optimal elimination trees, Tech. Report CS-88-13, The Pennsylvania State University (1988)
Schäffer, A.A.: Optimal node ranking of trees in linear time. Inform. Process. Lett. 33, 91–96 (1989-1990)
de la Torre, P., Greenlaw, R., Schäffer, A.A.: Optimal edge ranking of trees in polynomial time. Algorithmica 13, 529–618 (1995)
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Dereniowski, D., Kubale, M. (2004). Cholesky Factorization of Matrices in Parallel and Ranking of Graphs. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2003. Lecture Notes in Computer Science, vol 3019. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24669-5_127
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DOI: https://doi.org/10.1007/978-3-540-24669-5_127
Publisher Name: Springer, Berlin, Heidelberg
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