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.
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Notes
A graph is called friendly if its adjacency matrix has a simple spectrum and eigenvectors orthogonal to \(\mathbf {1}_n\) [1].
The code used in this work is available at https://github.com/fsbravo/csdp.git.
References
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
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)
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)
Babai, L.: Graph isomorphism in quasipolynomial time. ArXiv e-prints (2015)
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
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
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
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
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
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
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
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
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
Elias Oliveira, D., Wolkowicz, H., Xu, Y.: ADMM for the SDP relaxation of the QAP. ArXiv e-prints (2015)
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
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
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
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
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
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
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
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)
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
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
Ramana, M.V., Scheinerman, E.R., Ullman, D.: Fractional isomorphism of graphs. Discrete Math. 132(1–3), 247–265 (1994)
Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976). https://doi.org/10.1145/321958.321975.
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
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
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
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
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
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
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
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.
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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
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DOI: https://doi.org/10.1007/s10589-017-9968-8