Semidefinite programming approach for the quadratic assignment problem with a sparse graph

Abstract

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension \(n^2\) where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension \(\mathcal {O}(n)\) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension \(\mathcal {O}(n)\). The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Notes

  1. 1.

    A graph is called friendly if its adjacency matrix has a simple spectrum and eigenvectors orthogonal to \(\mathbf {1}_n\) [1].

  2. 2.

    The code used in this work is available at https://github.com/fsbravo/csdp.git.

References

  1. 1.

    Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proc. Natl. Acad. Sci. 112(10), 2942–2947 (2015). https://doi.org/10.1073/pnas.1401651112. http://www.pnas.org/content/112/10/2942.abstract

  2. 2.

    Alipanahi, B., Gao, X., Karakoc, E., Li, S., Balbach, F., Feng, G., Donaldson, L., Li, M.: Error tolerant nmr backbone resonance assignment and automated structure generation. J. Biomol. NMR 9(1), 15–41 (2011)

    Google Scholar 

  3. 3.

    Almohamad, H., Duffuaa, S.O.: A linear programming approach for the weighted graph matching problem. IEEE Trans. Pattern Anal. Mach. Intell. 15(5), 522–525 (1993)

    Article  Google Scholar 

  4. 4.

    Babai, L.: Graph isomorphism in quasipolynomial time. ArXiv e-prints (2015)

  5. 5.

    Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB–a quadratic assignment problemlibrary. J. Glob. Optim. 10(4), 391–403 (1997). https://doi.org/10.1023/A:1008293323270

    Article  MATH  Google Scholar 

  6. 6.

    Cavuslar, G., Catay, B., Apaydin, M.S.: A tabu search approach for the nmr protein structure-based assignment problem. IEEE/ACM Trans. Comput. Biol. Bioinform. 9(6), 1621–1628 (2012). https://doi.org/10.1109/TCBB.2012.122

    Article  Google Scholar 

  7. 7.

    Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of admm for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155(1–2), 57–79 (2016). https://doi.org/10.1007/s10107-014-0826-5

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chen, L., Sun, D., Toh, K.C.: An efficient inexact symmetric gauss–seidel based majorized admm for high-dimensional convex composite conic programming. Math. Program. 161(1–2), 237–270 (2017). https://doi.org/10.1007/s10107-016-1007-5

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Christofides, N., Benavent, E.: An exact algorithm for the quadratic assignment problem on a tree. Oper. Res. 37(5), 760–768 (1989). http://www.jstor.org/stable/171021

  10. 10.

    de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program. 122(2), 225–246 (2010). https://doi.org/10.1007/s10107-008-0246-5

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    de Klerk, E., Sotirov, R., Truetsch, U.: A new semidefinite programming relaxation for the quadratic assignment problem and its computational perspectives. INFORMS J. Comput. 27(2), 378–391 (2015). https://doi.org/10.1287/ijoc.2014.0634

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Dufoss, F., Uar, B.: Notes on birkhoffvon neumann decomposition of doubly stochastic matrices. Linear Algebra Appl. 497, 108–115 (2016). https://doi.org/10.1016/j.laa.2016.02.023. http://www.sciencedirect.com/science/article/pii/S0024379516001257

  13. 13.

    Eghbalnia, H.R., Bahrami, A., Wang, L., Assadi, A., Markley, J.L.: Probabilistic identification of spin systems and their assignments including coil-helix inference as output (pistachio). J. Biomol. NMR 32(3), 219–233 (2005). https://doi.org/10.1007/s10858-005-7944-6

    Article  Google Scholar 

  14. 14.

    Elias Oliveira, D., Wolkowicz, H., Xu, Y.: ADMM for the SDP relaxation of the QAP. ArXiv e-prints (2015)

  15. 15.

    Eschermann, B., Wunderlich, H.J.: Optimized synthesis of self-testable finite state machines. In: Fault-Tolerant Computing, 1990. FTCS-20. Digest of Papers., 20th International Symposium, pp. 390–397 (1990). https://doi.org/10.1109/FTCS.1990.89393

  16. 16.

    Genton, M.G.: Classes of kernels for machine learning: A statistics perspective. J. Mach. Learn. Res. 2, 299–312 (2002). http://dl.acm.org/citation.cfm?id=944790.944815

  17. 17.

    Jung, Y.S., Zweckstetter, M.: Mars—robust automatic backbone assignment of proteins. J. Biomol. NMR 30(1), 11–23 (2004). https://doi.org/10.1023/B:JNMR.0000042954.99056.ad

    Article  Google Scholar 

  18. 18.

    Kezurer, I., Kovalsky, S.Z., Basri, R., Lipman, Y.: Tight relaxation of quadratic matching. Comput. Gr. Forum (2015). https://doi.org/10.1111/cgf.12701

    Google Scholar 

  19. 19.

    Koopmans, T., Beckmann, M.J.: Assignment problems and the location of economic activities. Cowles Foundation Discussion Papers 4, Cowles Foundation for Research in Economics, Yale University (1955). http://EconPapers.repec.org/RePEc:cwl:cwldpp:4

  20. 20.

    Li, X., Sun, D., Toh, K.C.: A schur complement based semi-proximal admm for convex quadratic conic programming and extensions. Math. Program. 155(1–2), 333–373 (2016). https://doi.org/10.1007/s10107-014-0850-5

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Loiola, E.M., de Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P., Querido, T.: A survey for the quadratic assignment problem. Eur. J. Oper. Res. 176(2), 657–690 (2007). https://doi.org/10.1016/j.ejor.2005.09.032. http://www.sciencedirect.com/science/article/pii/S0377221705008337

  22. 22.

    Lyzinski, V., Fishkind, D.E., Fiori, M., Vogelstein, J.T., Priebe, C.E., Sapiro, G.: Graph matching: relax at your own risk. IEEE Trans. Pattern Anal. Mach. Intell. 38(1), 60 (2016)

    Article  Google Scholar 

  23. 23.

    Peng, J., Mittelmann, H., Li, X.: A new relaxation framework for quadratic assignment problems based on matrix splitting. Math. Program. Comput. 2(1), 59–77 (2010). https://doi.org/10.1007/s12532-010-0012-6

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Peng, J., Zhu, T., Luo, H., Toh, K.C.: Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting. Comput. Optim. Appl. 60(1), 171–198 (2015). https://doi.org/10.1007/s10589-014-9663-y

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Ramana, M.V., Scheinerman, E.R., Ullman, D.: Fractional isomorphism of graphs. Discrete Math. 132(1–3), 247–265 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976). https://doi.org/10.1145/321958.321975.

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Sturm, J.F.: Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999). https://doi.org/10.1080/10556789908805766

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Sun, D., Toh, K.C., Yang, L.: A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM J. Optim. 25(2), 882–915 (2015). https://doi.org/10.1137/140964357

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Ulrich, E.L., Akutsu, H., Doreleijers, J.F., Harano, Y., Ioannidis, Y.E., Lin, J., Livny, M., Mading, S., Maziuk, D., Miller, Z., Nakatani, E., Schulte, C.F., Tolmie, D.E., Kent Wenger, R., Yao, H., Markley, J.L.: BioMagResBank. Nucleic Acids Res. 36(suppl 1), D402–D408 (2008). https://doi.org/10.1093/nar/gkm957. http://nar.oxfordjournals.org/content/36/suppl_1/D402.abstract

  30. 30.

    Wan, X., Lin, G.: CISA: Combined NMR resonance connectivity information determination and sequential assignment. IEEE/ACM Trans. Comput. Biol. Bioinf. 4(3), 336–348 (2007). https://doi.org/10.1109/tcbb.2007.1047

    Article  Google Scholar 

  31. 31.

    Wuthrich, K., Wider, G., Wagner, G., Braun, W.: Sequential resonance assignments as a basis for determination of spatial protein structures by high resolution proton nuclear magnetic resonance. J. Mol. Biol. 155(3), 311–319 (1982). https://doi.org/10.1016/0022-2836(82)90007-9. http://www.sciencedirect.com/science/article/pii/0022283682900079

  32. 32.

    Zaslavskiy, M., Bach, F., Vert, J.P.: A path following algorithm for the graph matching problem. IEEE Trans. Pattern Anal. Mach. Intell. 31(12), 2227–2242 (2009). https://doi.org/10.1109/TPAMI.2008.245

    Article  Google Scholar 

  33. 33.

    Zhao, Q., Karisch, S., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2(1), 71–109 (1998). https://doi.org/10.1023/A:1009795911987

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank David Cowburn for useful discussions on NMR spectroscopy, and Amir Ali Ahmadi, for suggestions on tightening the SDP relaxations presented here. The authors would also like to thank the anonymous reviewers for their useful feedback and suggestions. The authors were partially supported by Award Number R01GM090200 from the NIGMS, FA9550-12-1-0317 from AFOSR, the Simons Investigator Award and the Simons Collaboration on Algorithms and Geometry from Simons Foundation, and the Moore Foundation Data-Driven Discovery Investigator Award.

Author information

Affiliations

Authors

Corresponding author

Correspondence to José F. S. Bravo Ferreira.

Appendix: full tabulated results

Appendix: full tabulated results

See Tables 3, 4, 5, 6 and Figs. 9 and 10.

Table 3 Comparison between lower bounds given by the C-SDP and Eigenspace relaxations on selected problems from the QAP library with (relatively) sparse B
Table 4 Comparison between upper bounds given by C-SDP and PATH relaxations on elected problems from the QAP library with (relatively) sparse B
Table 5 Comparison between lower bounds given by the C-SDP and Eigenspace relaxations on problems from the TSP library (with \(n\le 150\))
Table 6 Comparison between upper bounds given by C-SDP and PATH relaxations on problems from the TSP library (with \(n\le 150\))
Fig. 9
figure9

Comparison of run times between C-SDP and Eigenspace (linear program) relexations on problems from the TSP library (with \(n\le 150\)). C-SDP instances were ran for 1000 ADMM iterations on 20 processors. Eigenspace instances were solved using SeDuMi [27]. As the Eigenspace relaxation simplifies to a linear program in the case of TSP, solving the problem using an interior point solver is stil competitive with the ADMM approach used in C-SDP

Fig. 10
figure10

Comparison of run times between C-SDP and Eigenspace relexations on problems from the QAP library (with \(n\le 150\)). C-SDP instances were ran for 1000 ADMM iterations on 20 processors. Eigenspace instances were solved using SeDuMi [27]. In the case of problems from the QAP library, the Eigenspace relaxation no longer simplifies, resulting in longer runtimes

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bravo Ferreira, J.F.S., Khoo, Y. & Singer, A. Semidefinite programming approach for the quadratic assignment problem with a sparse graph. Comput Optim Appl 69, 677–712 (2018). https://doi.org/10.1007/s10589-017-9968-8

Download citation

Keywords

  • Graph matching
  • Quadratic assignment problem
  • Convex relaxation
  • Semidefinite programming
  • Alternating direction method of multipliers