Skip to main content
Springer Nature Link
Log in

Phase Transition in the Bandwidth Minimization Problem

  • Conference paper
MICAI 2009: Advances in Artificial Intelligence (MICAI 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5845))

Included in the following conference series:

  • 1594 Accesses

Abstract

It is known that some NP-Complete problems exhibit sharp phase transitions with respect to some order parameter. Moreover, a correlation between that critical behavior and the hardness of finding a solution exists in some of these problems. This paper shows experimental evidence about the existence of a critical behavior in the computational cost of solving the bandwidth minimization problem for graphs (BMPG). The experimental design involved the density of a graph as order parameter, 200000 random connected graphs of size 16 to 25 nodes, and a branch and bound algorithm taken from the literature. The results reveal a bimodal phase transition in the computational cost of solving the BMPG instances. This behavior was confirmed with the results obtained by metaheuristics that solve a known BMPG benchmark.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Karp, R.: On the computational complexity of combinatorial problems. Networks 5(1), 45–68 (1975)

    MATH  Google Scholar 

  2. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  3. Assman, S., Peck, G., Syslo, M., Zak, J.: The bandwidth of carterpillars with hairs of length 1 and 2. SIAM J. Algebraic and Discrete Methods 2, 387–393 (1981)

    Article  MathSciNet  Google Scholar 

  4. Monein, B.: The bandwidth minimization problem for carterpillars with hair length 3 is np-complete. SIAM J. Algebraic Discrete Methods 7(4), 505–512 (1986)

    Article  MathSciNet  Google Scholar 

  5. Taylor, W., Cheeseman, P., Kanefsky, B.: Where the really hard problems are. In: Mylopoulos, J., Reiter, R. (eds.) Proceedings of 12th International Joint Conference on AI (IJCAI 1991), vol. 1, pp. 331–337. Morgan Kauffman, San Francisco (1991)

    Google Scholar 

  6. Fu, Y., Anderson, P.: Application of statistical mechanics to np-complete problems in combinatorial optimisation. Journal of Physics A: Mathematical and General 19(9), 1605–1620 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Huberman, B., Hogg, T.: Phase transitions in artificial intelligence systems. Artif. Intell. 33(2), 155–171 (1987)

    Article  Google Scholar 

  8. Barbosa, V., Ferreira, R.: On the phase transitions of graph coloring and independent sets. Physica A 343, 401–423 (2004)

    Article  Google Scholar 

  9. Hartmann, A., Weigt, M.: Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs. Theoretical Computer Science 265, 199–225 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kirkpatrick, S., Selman, B.: Critical behavior in the satisfiability of random Boolean expressions. Artificial Intelligence 81, 273–295 (1994)

    MathSciNet  Google Scholar 

  11. Smith, B.: Constructing an asymptotic phase transition in random binary constraint satisfaction problems. Theoretical Computer Science 265, 265–283 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mazure, B., Sais, L., Grégoire, E.: Tabu search for sat. In: Proceedings of the Fourteenth Natĺ Conf. on Artificial Intelligence (AAAI 1997), pp. 281–285. AAAI Press, Menlo Park (1997)

    Google Scholar 

  13. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic ṕhase transitions.́ Nature 400, 133–137 (1999)

    Article  MathSciNet  Google Scholar 

  14. Navarro, J.A., Voronkov, A.: Generation of hard non-clausal random satisfiability problems. In: AAAI 2005 (2005)

    Google Scholar 

  15. Herroelen, W., De Reyck, B.: Phase transitions in project scheduling. Journal of the Operational Research Society 50, 148–156 (1999)

    Article  MATH  Google Scholar 

  16. Slaney, J., Walsh, T.: Phase transition behavior: from decision to optimization. In: Proceedings of the 5th International Symposium on the Theory and Applications of Satisfiability Testing, SAT (2002)

    Google Scholar 

  17. Achlioptas, D., Naor, Y.P. A.: Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005)

    Article  Google Scholar 

  18. Gent, I.P., Walsh, T.: Phase transitions and annealed theories: Number partitioning as a case study. In: ECAI, pp. 170–174 (1996)

    Google Scholar 

  19. Gent, I., Walsh, T.: Analysis of heuristics for number partitioning. Computational Intelligence 14(3), 430–451 (1998)

    Article  MathSciNet  Google Scholar 

  20. Caprara, A., Salazar-González, J.: Laying out sparse graphs with provably minimum bandwidth. Informs Journal on Computing 17(3), 356–373 (2005)

    Article  MathSciNet  Google Scholar 

  21. Cuthill, E.H., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proc. 24th ACM National Conf. pp. 157–172 (1969)

    Google Scholar 

  22. Esposito, A., Catalano, M., Malucelli, F., Tarricone, L.: Sparse matrix bandwidth reduction: Algorithms, applications and real industrial cases in electromagnetics. Advances in the Theory of Computation and Computational Mathematics 2, 27–45 (1998)

    Google Scholar 

  23. Bhatt, S., Leighton, F.: A framework for solving VLSI graph layout problems. J. Comput. Sytem Sci. 28, 300–343 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. Berry, M., Hendrickson, B., Raghavan, P.: Sparse matrix reordering schemes for browsing hypertext. Lectures in Applied Mathematics 32, 99–123 (1996)

    Google Scholar 

  25. Papadimitriou, C.: The np-completeness of the bandwidth minimization problem. Computing 16(3), 263–270 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  26. Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics 34(3), 477–495 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  27. Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is np-complete. SIAM J. Algebraic Discrete Methods 7(4), 505–512 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  28. Blache, G., Karpinski, M., Wirtgen, J.: On approximation intractability of the bandwidth problem. Electronic Colloquium on Computational Complexity (ECCC) 5(014) (1998)

    Google Scholar 

  29. Del Corso, G.M., ManZini, G.: Finding exact solutions to the bandwidth minimization problem. Computing 62, 189–203 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Marti, R., Campos, V., Pinana, E.: A branch and bound algorithm for the matrix bandwidth minimization. EJORS 186, 513–528 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pinana, E., Plana, I., Campos, V., Marti, V.: GRASP and path relinking for the matrix bandwdith minimization. European Journal of Operational Research 153, 200–210 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Marti, R., Laguna, M., Glover, F., Campos, V.: Reducing the bandwidth of a sparse matrix with tabu search. European Journal of Operational Research 135(2), 211–220 (2001)

    Article  MathSciNet  Google Scholar 

  33. Rodriguez-Tello, E., Hao, J., Torres-Jimenez, J.: An improved simulated annealing algorithm for bandwidth minimization. European Journal of Operational Research 185(3), 1319–1335 (2008)

    Article  MATH  Google Scholar 

  34. Erdos, P., Rényi, A.: On random graphs. i. Publicationes Mathematicae 6, 290–297 (1959)

    Google Scholar 

  35. Prüffer, P.: Neuer beweis eines satzes über permutationen. Arch. Math. Phys. 27, 742–744 (1918)

    Google Scholar 

  36. Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence 81, 273–295 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Technology Laboratory, CINVESTAV-Tamaulipas, Km. 6 Carretera Victoria-Monterrey, 87276, Victoria Tamps., Mexico

    Nelson Rangel-Valdez & Jose Torres-Jimenez

Authors
  1. Nelson Rangel-Valdez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jose Torres-Jimenez
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centro de Investigación en Matemáticas, AC. (CIMAT), Area de Computación, Calle Jalisco S/N, Mineral de Valenciana, Guanajuato, CP 36250, Guanajuato, México

    Arturo Hernández Aguirre

  2. Ciencias de la Computación, Tecnológico de Monterrey, Campus Estado de México, Carretera al lago de Guadalupe Km 3,5, Col. Margarita Maza de Juárez, Atizapán de Zaragoza,, CP 52926, Estado de México, México

    Raúl Monroy Borja

  3. Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE), Ciencias Computacionales, Luis Enrique Erro No. 1, Santa María Tonantzintla, CP 72840, Puebla, México

    Carlos Alberto Reyes Garciá

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rangel-Valdez, N., Torres-Jimenez, J. (2009). Phase Transition in the Bandwidth Minimization Problem. In: Aguirre, A.H., Borja, R.M., Garciá, C.A.R. (eds) MICAI 2009: Advances in Artificial Intelligence. MICAI 2009. Lecture Notes in Computer Science(), vol 5845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05258-3_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-05258-3_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05257-6

  • Online ISBN: 978-3-642-05258-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics