Carlos J. Luz *
* Escola Superior de Tecnologia de Setúbal / Instituto Politécnico de Setúbal cluz@est.ips.pt
Abstract
A method of determining a lower bound for the graph bisection minimization problem is described. The bound is valid for weigthed graphs with edge and node weights. The approach is based on Lagrangian relaxation and was previously used for determining an upper bound on the independence number of a graph. The determination of the lower bound is done by solving a quadratic programming problem. A characterization of the solutions of this problem is proved which allows to approximate the optimal solution of the graph bisection minimization problem. Some computational experiments are reported.
Keywords: Combinatorial Optimization, Graph Bisection Problem, Quadratic Programming
Texto completo disponível apenas em PDF.
]]> Full text only available in PDF format.
Bibliography [ Links ]
2. Boppana, R. B., Eigenvalues and graph bisection: an average case analysis, in Proc. 28th Annual Symposium on Computer Science, IEEE, 280-285, 1987. [ Links ]
3. Donath, W. and Hoffman, A., Algorithms for partitioning of graphs and computer logic based on eigenvectors of connection matrices, IBM Technical Disclosure Bulletin, 15, 398-944, 1972. [ Links ]
4. Donath, W. and Hoffman, A., Lower bounds for the partitioning of graphs, IBM J. Res. Develop., 17, 420-425, 1973. [ Links ]
5. Dias, O. P., Teixeira, I. C., and Teixeira, J. P., Metrics and criteria for quality assessment of testable hw/sw systems architectures, Journal of Electronic Testing: Theory and Applications (JETTA), 4(1), 149-158, 1999. [ Links ]
6. Falkner, J., Rendl, F. and Wolkowicz, H., A computational study of graph partitioning, Math. Programming, 66, 211-240, 1994. [ Links ]
7. Fiedler, M., Algebric connectivity of graphs, Czechoslovak Math. J., 23, pp. 298-305, 1973. [ Links ]
8. Fiedler, M., A property of eigenvectors of nonnegative symmetric matrices and its application to graph, Czechoslovak Math. J., 25, pp. 619-633, 1975. [ Links ]
9. Fiduccia, C. M. and Mattheyses, R. M., A linear-time heuristic for improving networks partitions, Proceedings of the 19th IEEE Designs Automation Conference, 175-181, 1982. [ Links ]
10. Frieze, A. and Jerrum, M., Improved approximation algorithms for max k-cut and max bisection, Proc. 4th IPCO Conference, 1-13, 1995. [ Links ]
11. Goemans, M. X, Semidefinite programming in combinatorial optimization, Mathematical Programming, 79, 143-161, 1997. [ Links ]
12. Goemans, M. X. and Williamson, D. P, Improved approximation algorithms for maximum Cut and satisfiability problems using semidefinite programming, Journal of ACM, 42, 1115-1145, 1995. [ Links ]
13. Hendrickson, B. and Leland, R., A Multilevel algorithm for partitioning graphs, Technical Report SAND93-1301, Sandia National Laboratories, 1993. [ Links ]
14. Hendrickson, B. e Leland, R., The Chaco user's guide: Version 2, Tecnhical Report SAND94-2692, Sandia National Laboratories, Albuquerque, NM, 1994. [ Links ]
15. Hager, W. W. and Krylyuk, Y., Graph partitioning and continuous quadratic programming, SIAM J. Discrete Math, 12(4), 500-523, 1999. [ Links ]
16. Hendrickson, B. and Leland, R., An improved spectral graph partitionning algorithm for mapping parallel computations, SIAM, 16, n° 2, pp. 452-469, 1995. [ Links ]
17. Horst, R., Pardalos, P. M., and Thoai, N. V., Introduction to Global Optimization, Kluwer Academic Publishers, 1995. [ Links ]
18. Karypis, G. ans Kumar, V., Multilevel k-way partitioning scheme for irregular graphs, 1995. [ Links ]
19. Kernighan, B. and Lin, S., An efficient heuristic procedur for partitioning graphs, Bell System Technical Journal, 29, pp. 291-307, 1970. [ Links ]
20. Lengauer, T., Combinatorial algorithms for integrated circuit layout, John Wiley and Sons, Chicester, 1990. [ Links ]
21. Luz, C. J. , An upper bound on the independence number of a graph, Operations Research Letters, 18, 139-145, 1995. [ Links ]
22. Luz, C. J. and Cardoso, D. M., A generalization of the Hoffman-Lovász upper bound on the independence number of a regular graph, Annals of Operations Research, 81, 307-319, 1998. [ Links ]
23. Peressini, A. L., Sullivan, F. E. and Uhl, J. J., The Mathematics of Nonlinear Programming, Springer, Berlin, 1993. [ Links ]
24. Poljak, S. and Rendl, F., Nonpolyhedral relaxations of graph-bisection problems, SIAM J. Optimization, 5(3), 467-487, 1995. [ Links ]
25. Pothen, A., Simon, H. D., and Liou, K.-P., Partitioning sparse matrices with eigenvectors of graphs, SIAM J. Matrix Anal. Appl., 11, pp.430-452, 1990. [ Links ]
26. Powers, D., Graph partitioning by eigenvectors, Lin. Alg. Appl., 101, 121-133, 1988. [ Links ]
27. Rendl, F. and Wolkowicz, H, A projection technique for partitioning the nodes of a graph, Tech. Rep. CORR 90-20, University of Waterloo, Faculty of Mathematics, Waterloo, Ontario, 1990. [ Links ]
28. Ye, Y., A .699-approximation algorithm for max-bisection, Research Report, University of Iowa, 2000. [ Links ] ]]>