Abstract
Decomposition of large engineering system models is desirable sinceincreased model size reduces reliability and speed of numericalsolution algorithms. The article presents a methodology for optimalmodel-based decomposition (OMBD) of design problems, whether or notinitially cast as optimization problems. The overall model isrepresented by a hypergraph and is optimally partitioned into weaklyconnected subgraphs that satisfy decomposition constraints. Spectralgraph-partitioning methods together with iterative improvementtechniques are proposed for hypergraph partitioning. A known spectralK-partitioning formulation, which accounts for partition sizes andedge weights, is extended to graphs with also vertex weights. TheOMBD formulation is robust enough to account for computationaldemands and resources and strength of interdependencies between thecomputational modules contained in the model.
Similar content being viewed by others
References
C.J. Alpert and A.B. Kahng, "Geometric embeddings for faster and better multi-way netlist partitioning," in Proceedings 30th ACM/IEEE Design Automation Conference, pp. 743–748, 1993.
C.J. Alpert and S.-Z. Yao, "Spectral partitioning: The more eigenvectors, the better," in Proceedings 32nd ACM/IEEE Design Automation Conference, June 1995.
S.T. Barnard and H.D. Simon, "A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems," in Proceedings 6th SIAM Conference on Parallel Processing for Scientific Computing, pp. 711–718, 1993.
E. Barnes, "An algorithm for partitioning the nodes of a graph," SIAM Journal on Algebrauc Discrete Methods, vol. 3, no. 4, pp. 541–550, 1982.
E. Barnes, A. Vannelli, and J. Walker, "A new heuristic for partitioning the nodes of a graph," SIAM Journal on Discrete Mathematics, vol. 1, no. 3, pp. 299–305, 1988.
C.L. Bloebaum, "Formal and heuristic system decomposition methods in multidisciplinary synthesis," NASA Contractor Report 4413, 1991.
M. Bolla, "Spectra, euclidean representations and clusterings of hypergraphs," Discrete Mathematics, vol. 117, pp. 19–39, 1993.
R.B. Boppana, "Eigenvalues and graph bisection: An average-case analysis," in Proceedings 28th IEEE Symposium on Foundations of Computer Science, pp. 280–285, 1987.
T.N. Bui, S. Chauduri, F.T. Leighton, and M. Sipser, "Graph bisection algorithms with good average case behavior," Combinatorica, vol. 7, no. 2, pp. 171–191, 1987.
T.N. Bui, C. Heigham, C. Jones, and T. Leighton, "Improving the performance of the Kernighan-Lin and simulated annealing graph bisection algorithms," in Proceedings 26th ACM/IEEE Design Automation Conference, pp. 775–778, 1989.
T.N. Bui and B.R. Moon, "A fast and stable hybrid genetic algorithm for the ratio-cut partitioning problem on hypergraphs," in Proceedings 31th ACM/IEEE Design Automation Conference, pp. 664–669, 1994.
P.K. Chan, M.D.F. Schlag, and J.Y. Zien, "Spectral K-way ratio-cut partitioning and clustering," IEEE Transactions on Computer-Auded Design of Integrated Circuits and Systems, vol. 13, no. 9, pp. 1088–1096, 1994.
R.D. Consoli and J. Sobieszczanski-Sobieski, "Application of advanced multidisciplinary analysis and optimization methods to vehicle design synthesis," Journal of Aurcraft, vol. 29, no. 5, pp. 811–818, 1992.
J.K. Cullum and R.A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Birkh¨auser: Boston, vol. 1, 1985.
W.E. Donath, "Logic partitioning," Physical Design Automation of VLSI Systems, B.T. Preas and M.J. Lorenzetti (Eds.), Benjamin Cummings: Menlo Park, California, Chap. 3, 1988.
W.E. Donath and A.J. Hoffman, "Lower bounds for the partitioning of graphs," IBM Journal of Research and Development, vol. 1, pp. 420–425, 1973.
I.S. Duff, A.M. Erisman, and J.K. Reid, Direct Methods for Sparse Matrices, Oxford Science Publications: Oxford, 1989.
A. Dunlop and B.W. Kernighan, "A procedure for placement of standard-cell VLSI circuits," IEEE Transactions on Computer-Auded Design, vol. CAD-4, no. 1, pp. 92–98, 1985.
S.D. Eppinger, D.E. Whitney, and D.A. Gebala, "Organizing the tasks in complex design projects: Development of tools to represent design procedures," in Proceedings NSF Design and Manufacturing System Conference, Atlanta, Georgia, Jan. 1992.
S.D. Eppinger, D.E. Whitney, R.P. Smith, and D.A. Gebala, "A model-based method for organizing tasks in product development," Research in Engineering Design, vol. 6, pp. 1–13, 1994.
J. Falkner, F. Rendl, and H. Wolkowicz, "A computational study of graph partitioning," Mathematical Programming, vol. 66, pp. 211–239, 1994.
C. Farhat and M. Lesoinne, "Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics," International Journal for Numerical Methods in Engineering, vol. 36, pp. 745–764, 1993.
D. Ferrari, "Improving locality by critical working sets," Communications of the ACM, vol. 17, no. 11, pp. 614–620, 1974.
C.M. Fiduccia and R.M. Mattheyses, "Alinear-time heuristic for improving network partitions," in Proceedings 19th ACM/IEEE Design Automation Conference, pp. 175–181, 1982.
M. Fiedler, "Algebrauc connectivity of graphs," Czechoslovak Mathematics Journal, vol. 23, pp. 298–305, 1973.
M. Fiedler, "Laplacian of graphs and algebrauc connectivity," Combinatorics and Graph Theory, Banach Center Publications: Warsaw, vol. 25, pp. 57–70, 1989.
O. Goldschmidt and D. Hochbaum, "A polynomial algorithm for the k-Cut problem for fixed k," Mathematics of Operations Research, vol. 19, no. 1, pp. 24–37, 1994.
S. Hadley, B. Mark, and A. Vannelli, "An efficient Eigenvector approach for finding netlist partitions," IEEE Transactions on Computer-Auded Design, vol. 11, no. 7, pp. 885–892, 1992.
S. Hadley, F. Rendl, and H. Wolkowicz, "A new lower bound via projection for the quadratic assignment problem," Mathematics of Operations Research, vol. 17, no. 3, pp. 727–739, 1992.
Y. Haumes, K. Tarvaunen, T. Shima, and J. Thadathil, Hierarchical Multiobjective Analysis of Large-Scale Systems, HPC: New York, 1990.
B. Hendrickson and R. Leland, "An improved spectral graph partitioning algorithm for mapping parallel computations," SIAM Journal on Scientific Computing, vol. 16, no. 2, pp. 452–469, 1995.
B. Hendrickson and R. Leland, "The Chaco user's guide version 2.0," Technical Report SAND94-2692, Sandia National Labs, Albuquerque, New Mexico, 1995.
W. Hock and K. Schittkowski, "Test examples for nonlinear programming codes," Lecture Notes in Economics and Mathematical Systems, no. 187, M. Beckman and H.P. Künzi (Eds.), Springer-Verlag: Berlin, 1981.
D.S. Johnson, C.R. Aragon, L.A. McGeoch, and C. Schevon, "Optimization by simulated annealing: An experimental evaluation; part I, graph partitioning," Operations Research, vol. 37, no. 6, pp. 865–891, 1989.
B.W. Kernighan and S. Lin, "An efficient heuristic procedure for partitioning graphs," The Bell System Technical Journal, vol. 49, pp. 291–307, Feb. 1970.
B. Krishnamurthy, "An improved min-cut algorithm for partitioning VLSI networks," IEEE Transactions on Computers, vol. C-33, no. 5, pp. 438–446, 1984.
A. Kusiak and J. Wang, "Efficient organizing of design activities," International Journal of Production Research, vol. 31, no. 4, pp. 753–769, 1993.
A. Kusiak and J. Wang, "Decomposition of the design process," trans. ASME, Journal of Mechanical Design, vol. 115, pp. 687–695, Dec. 1993.
T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, John Wiley & Sons: Chichester, 1990.
J. MacQueen, "Some methods for classification and analysis of multivariate observations," in Proceedings 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297, 1967.
N. Michelena and P. Papalambros, "A network reliability approach to optimal decomposition of design problems," Trans. ASME, Journal of Mechanical Design, vol. 117, no. 3, pp. 433–440, Sept. 1995.
N. Michelena and P. Papalambros, "Optimal model-based partitioning of powertraun system design," Trans. ASME, Journal of Mechanical Design, vol. 117, no. 4, pp. 499–505, Dec. 1995.
B. Mohar, "The Laplacian spectrum of graphs," in Proceedings 6th International Conference on Theory and Applications of Graphs, Kalamazoo, Michigan, 1988, vol. 2, pp. 871–898.
B. Mohar, "Laplace eigenvalues of graphs-A survey," Discrete Mathematics, vol. 109, pp. 171–183, 1992.
B.N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Inc.: Englewood Cliffs, 1980.
A. Pothen, H. Simon, and K. Liou, "Partitioning sparse matrices with eigenvectors of graphs," SIAM Journal on Matrix Analysis and Applications, vol. 11, no. 3, pp. 430–452, 1990.
F. Rendl and H. Wolkowicz, "A projection technique for partitioning the nodes of a graph," Research Report CORR 90–20, University of Waterloo, Canada, 1990.
F. Rendl and H. Wolkowicz, "Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem," Mathematical Programming, vol. 53, pp. 63–78, 1992.
B. Riess, K. Doll, and F. Johannes, "Partitioning very large circuits using analytical placement techniques," in Proceedings 31st ACM/IEEE Design Automation Conference, pp. 646–651, 1994.
J.L. Rogers and C.L. Bloebaum, "Ordering design tasks based on coupling strengths," Proceedings 5th AUAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Panama City, 1994, AUAA paper 94-4326-CP, pp. 708–717.
L.A. Sanchis, "Multiple-way network partitioning," IEEE Transactions on Computers, vol. 38, no. 1, pp. 62–81, 1989.
L.A. Sanchis, "Multiple-way network partitioning with different cost functions," IEEE Transactions on Computers, vol. 42, no. 12, pp. 1500–1504, 1993.
H. Simon, "Partitioning of unstructured problems for parallel processing," Computing Systems in Engineering, vol. 2, no. 2/3, pp. 135–148, 1991.
J. Sobieszczanski-Sobieski, "Optimization by decomposition: Step from hierarchic to non-hierarchic systems," NASA TM 101494, Langley Research Center, Hampton, Virginia, 1988.
J. Sobieszczanski-Sobieski, "Sensitivity of complex, internally coupled systems," AUAA Journal, vol. 28, pp. 153–160, Jan. 1990.
J. Sobieszczanski-Sobieski, "Optimization by decomposition," Structural Optimization: Status and Promise, M.P. Kamat (Ed.), Progress in Astronautics and Aeronautics Series, vol. 150, pp. 487–515, 1993.
D.V. Steward, Systems Analysis and Management: Structure, Strategy, and Design, Petrocelli Books, Inc.: New York, 1981.
D.V. Steward, "The design structure system: A method for managing the design of complex systems," IEEE Transactions on Engineering Management, vol. EM-28, no. 3, pp. 71–74, 1981.
P. Suaris and G. Kedem, "An algorithm for quadrisection and its application to standard cell placement," IEEE Transactions on Circuits and Systems, vol. 35, no. 3, pp. 294–303, 1988.
T. Tokuyama and J. Nakano, "Geometric algorithms for a minimum cost assignment problem," in Proceedings 7th ACM Symposium on Computational Geometry, pp. 262–271, 1991.
N. Tzannetakis, M. Jensen, and J. Novak, "Development of optimal design methodologies for engine aur management," presented at the 15th International Symposium on Mathematical Programming, Ann Arbor, Michigan, 1994.
A.J. Vakharia, "Methods of cell formation in group technology: A framework for evaluation," International Journal of Operations Management, vol. 6, no. 3, pp. 257–271, 1986.
A. Vannelli and K. Kumar, "A method for finding minimal bottleneck cells for grouping part-machine families," International Journal of Production Research, vol. 24, pp. 387–400, 1986.
T.C. Wagner, A General Decomposition Methodology for Optimal System Design, Ph.D. dissertation, Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, Michigan, 1993.
T.C. Wagner and P.Y. Papalambros, "A general framework for decomposition analysis in optimal design," Advances in Design Automation-1993, B.J. Gilmore (Ed.), ASME: New York, vol. 2, pp. 315–325, 1993.
T.C. Wagner and P.Y. Papalambros, "Implementation of decomposition analysis in optimal design," Advances in Design Automation-1993, B.J. Gilmore (Ed.), ASME: New York, vol. 2, pp. 327–335, 1993.
Y.C. Wei and C.K. Cheng, "Ratio cut partitioning for hierarchical designs," IEEE Transactions on Computer-Auded Design, vol. 10, no. 7, pp. 911–921, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Michelena, N.F., Papalambros, P.Y. A Hypergraph Framework for Optimal Model-Based Decomposition of Design Problems. Computational Optimization and Applications 8, 173–196 (1997). https://doi.org/10.1023/A:1008673321406
Issue Date:
DOI: https://doi.org/10.1023/A:1008673321406