Semidefinite Programming for Graph Partitioning with Preferences in Data Distribution

  • Suely Oliveira
  • David Stewart
  • Takako Soma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2565)


Graph partitioning with preferences is one of the data distribution models for parallel computer, where partitioning and mapping are generatedto gether. It improves the overall throughput of message trafic by having communication restrictedto processors which are near each other, whenever possible. This model is obtained by associating to each vertex a value which reflects its net preference for being in one partition or another of the recursive bisection process. We have formulated a semidefinite programming relaxation for graph partitioning with preferences andimp lemented efficient subspace algorithm for this model. We numerically comparedou r new algorithm with a standardsem idefinite programming algorithm andsh ow that our subspace algorithm performs better.


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  1. [1]
    F. Alizadeh. Interior-point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1):13–51, 1995. 705MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Alizadeh, J. P. A. Haeberly, M. V. Nayakkankuppam, M. L. Overton, and S. Schmieta. SDPpack user’s guide-version 0.9 beta for Matlab 5.0. Technical Report TR1997-737, Computer Science Department, New York University, New York, NY, June 1997. 705Google Scholar
  3. [3]
    W. E. Arnoldi. The principle of minimized iteration in the solution of the matrix eigenproblem. Quarterly Applied Mathematics, 9:17–29, 1951. 709MathSciNetMATHGoogle Scholar
  4. [4]
    B. Borchers. CSDP, 2.3 User’s Guide. Optimization Methods and Software, 11(1):597–611, 1999. 705, 712CrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Crouzeix, B. Philippe, and M. Sadkane. The Davidson method. SIAM J. Sci. Comput., 15(1):62–76, 1994. 709MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    K. Fujisawa, M. Kojima, and K. Nakata. SDPA User’s Manual-Version 4.50. Technical Report B, Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan, July 1999. 705Google Scholar
  7. [7]
    M. R. Carey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete problems. Theoretical Computer Science, 1:237–267, 1976. 704CrossRefMathSciNetGoogle Scholar
  8. [8]
    B. Hendrickson. Graph partitioning and parallel solvers: Has the emperor no clothes? (extended abstract). In Lecture Notes in Computer Science, volume 1457, 1998. 704Google Scholar
  9. [9]
    B. Hendrickson and T. G. Kolda. Graph partitioning models for parallel computing. Parallel Comput., 26(12):1519–1534, 2000. 704MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. Hendrickson and R. Leland. The Chaco user’s guide, version 2.0. Technical Report SAND-95-2344, Sandia National Laboratories, Albuquerque, NM, July 1995. 705Google Scholar
  11. [11]
    B. Hendrickson, R. Leland, and R. Van Driessche. Enhancing data locality by using terminal propagation. In Proc. 29th Hawaii Intl. Conf. System Science, volume 16, 1996. 704Google Scholar
  12. [12]
    M. Holzrichter and S. Oliveira. A graph basedD avidson algorithm for the graph partitioning problem. International Journal of Foundations of Computer Science, 10:225–246, 1999. 705, 709CrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Holzrichter and S. Oliveira. A graph basedm ethodfor generating the Fiedler vector of irregular problems. In Lecture Notes in Computer Science, volume 1586, pages 978–985. Springer, 1999. Proceedings of the 11th IPPS/SPDP’99 workshops. 705, 709Google Scholar
  14. [14]
    S. E. Karisch and F. Rendl. Semidefinite programming and graph equipartition. In P. M. Pardalos and H. Wolkowicz, editors, Topics in Semidefinite and InteriorPoint Methods, volume 18, pages 77–95. AMS, 1998. 705Google Scholar
  15. [15]
    G. Karypis and V. Kumar. METIS: Unstructured graph partitioning andsp arse matrix ordering system Version 2.0. Technical report, Department of Computer Science, University of Minnesota, Minneapolis, MN, August 1995. 705Google Scholar
  16. [16]
    C. Lanczos. Solution of systems of linear equations by minimizedit erations. J. Research Nat’l Bureau of Standards, 49:33–53, 1952. 709MathSciNetGoogle Scholar
  17. [17]
    S. Oliveira. On the convergence rate of a preconditioned subspace eigensolver. Computing, 63(2):219–231, December 1999. 709, 710MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    S. Oliveira and T. Soma. A multilevel algorithm for spectral partitioning with extended eigen-models. In Lecture Notes in Computer Science, volume 1800, pages 477–484. Springer, 2000. Proceedings of the 15th IPDPS 2000 workshops. 704, 706Google Scholar
  19. [19]
    S. Oliveira, D. Stewart, and T. Soma. A subspace semidefinite programming for spectral graph partit ioning. In P.M.A Sloot, C.K.K. Tan, J.J. Dongarra, and A. G. Hoekstra, editors, Lecture Notes in Computer Science, volume 2329, pages 10581067. Springer, 2002. Proceedings of International Conference on Computational Science-ICCS 2002, Part 1, Amsterdam, The Netherlands. 704, 709Google Scholar
  20. [20]
    F. Pellegrini. SCOTCH 3.1 user’s guide. Technical Report 1137-96, Laboratoire Bordelais de Recherche en Informatique, Universite Bordeaux, France, 1996. 705Google Scholar
  21. [21]
    A. Pothen, H. D. Simon, and Kang-Pu K. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl., 11(3):430–452, 1990. 704, 705MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    R. Preis and R. Diekmann. The PARTY Partitioning-Library, User Guide-Version 1.1. Technical Report tr-rsfb-96-024, University of Paderborn, Germany, 1996. 705Google Scholar
  23. [23]
    F. Rendl. A Matlab toolbox for semidefinite programming. Technical report, Technische Universitdt Graz, Institut fir Mathematik, Kopernikusgasse 24, A-8010 Graz, Austria, 1994. 705Google Scholar
  24. [24]
    F. Rendl, R. J. Vanderbei, and H. Wolkowicz. primal-dual interior point algorithms, and trust region subproblems. Optimization Methods and Software, 5:1–16, 1995. 705CrossRefMathSciNetGoogle Scholar
  25. [25]
    Y. Saad. Numerical Methods for Large Eigenvalue Problems. Manchester University Press, OxfordR oad, Manchester M13 9PL, UK, 1992. 709Google Scholar
  26. [26]
    G. L. G. Sleijpen and H. A. Van der Vorst. A Jacobi-Davidson iteration method or linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17(2):401–425, 1996. Max-min eigenvalue problems, 709MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    D. E. Stewart and Z. Leyk. Meschach: Matrix Computations in C. Australian National University, Canberra, 1994. Proceedings of the CMA, # 32. 712Google Scholar
  28. [28]
    K. C. Toh, M. J. Todd, and P. H. Tiitüncii. SDPT3-a Matlab software package for semidefinite programming, version 2.1. Technical report, School of Operations Research andInd ustrial Engineering, Cornell University, Ithaca, NY, September 1999. 705Google Scholar
  29. [29]
    L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38:49–95, 1996. 705MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Vandenberghe and S. Boyd. SP Software for semidefinite programming User’s guide, version 1.0. Technical report, Information System Laboratory, Stanford University, Stanford, CA, November 1998. 705Google Scholar
  31. [31]
    C. Walshaw, M. Cross, and M. Everett. Mesh partitioning and load-balancing for distributed memory parallel systems. In B. Topping, editor, Proc. Parallel & Distributed Computing for Computational Mechanics, Lochinver, Scotland, 1998. 705Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Suely Oliveira
    • 1
  • David Stewart
    • 2
  • Takako Soma
    • 1
  1. 1.The Department of Computer ScienceThe University of IowaIowa CityUSA
  2. 2.The Department of MathematicsThe University of IowaIowa CityUSA

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