Skip to main content
SpringerLink
Log in

Building an iterative heuristic solver for a quantum annealer

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A quantum annealer heuristically minimizes quadratic unconstrained binary optimization (QUBO) problems, but is limited by the physical hardware in the size and density of the problems it can handle. We have developed a meta-heuristic solver that utilizes D-Wave Systems’ quantum annealer (or any other QUBO problem optimizer) to solve larger or denser problems, by iteratively solving subproblems, while keeping the rest of the variables fixed. We present our algorithm, several variants, and the results for the optimization of standard QUBO problem instances from OR-Library of sizes 500 and 2500 as well as the Palubeckis instances of sizes 3000–7000. For practical use of the solver, we show the dependence of the time to best solution on the desired gap to the best known solution. In addition, we study the dependence of the gap and the time to best solution on the size of the problems solved by the underlying optimizer. Our results were obtained by simulation, using a tabu 1-opt solver, due to the huge number of runs required and limited quantum annealer time availability.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The problem that the machine actually solves is known as an Ising model in the physics community, and can easily be transformed into the form \(x^T Q x\), which is more common in the scientific computing community.

  2. The chip is current as of April 2016.

  3. A one-flip gain is the change in the objective function’s value given a flip of a single bit. We refer to the collection of all possible single flips and their corresponding change in the objection function’s value as the “one-flip gains”.

  4. We chose 0.02 s based on 1000 anneals each taking 20  \({\upmu }\)s (which is the current minimum anneal time). This is the actual time for computation, and does not include extra time for programming the chip, thermalization, etc. Since future run times are not known, this number is only meant as a rough estimate, to give an idea of the actual time the computation could take.

  5. The source code of the generator and the input files to create these problems can be found at http://www.proin.ktu.lt/~gintaras/ubqop_its.html. This page was last retrieved on July 21, 2015.

References

  1. Boros, E., Prékopa, A.: Probabilistic bounds and algorithms for the maximum satisfiability problem. Ann. Oper. Res. 21(1–4), 109–126 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discrete Appl. Math. 123(1–3) pp. 155–225 (2002). Workshop on Discrete Optimization, DO’99, Piscataway

  3. Bourjolly, J.-M.: A quadratic 0-1 optimization algorithm for the maximum clique and stable set problems. Technical Report University of Michigan, Ann Arbor (1994)

  4. Du, D.-Z., Pardalos, P.M.: Handbook of Combinatorial Optimization: Supplement, vol. 1. Springer, New York (1999)

    Book  MATH  Google Scholar 

  5. Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2), 131–144 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pardalos, P.M., Rodgers, G.P.: A branch and bound algorithm for the maximum clique problem. Comput. Oper. Res. 19(5), 363–375 (1992)

    Article  MATH  Google Scholar 

  7. Pardalos, P.M., Xue, J.: The maximum clique problem. J. Global Optim. 4(3), 301–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kochenberger, G.A., Glover, F., Alidaee, B., Rego, C.: A unified modeling and solution framework for combinatorial optimization problems. OR Spect. 26(2), 237–250 (2004)

    Article  MATH  Google Scholar 

  9. Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36(3), 493–513 (1988)

    Article  MATH  Google Scholar 

  10. De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of ising spin glasses: New experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80(1–2), 487–496 (1995)

    Article  MATH  Google Scholar 

  11. Alidaee, B., Kochenberger, G.A., Ahmadian, A.: 0–1 quadratic programming approach for optimum solutions of two scheduling problems. Int. J. Syst. Sci. 25(2), 401–408 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. Handbook of Combinatorial Optimization, pp. 1–74. Springer, New York (1999)

    Chapter  Google Scholar 

  13. Iasemidis, L.D., Pardalos, P., Sackellares, J.C., Shiau, D.-S.: Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Comb. Optim. 5(1), 9–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Alidaee, B., Glover, F., Kochenberger, G.A., Rego, C.: A new modeling and solution approach for the number partitioning problem. J. Appl. Math. Decis. Sci. 2005(2), 113–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gulati, V., Gupta, S., Mittal, A.: Unconstrained quadratic bivalent programming problem. Eur. J. Oper. Res. 15(1), 121–125 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carter, M.W.: The indefinite zero-one quadratic problem. Discr. Appl. Math. 7(1), 23–44 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Williams, H.P.: Model Building in Linear and Integer Programming. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  18. Barahona, F., Jünger, M., Reinelt, G.: Experiments in quadratic 0–1 programming. Math. Prog. 44(1–3), 127–137 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pardalos, P.M., Rodgers, G.P.: Parallel branch and bound algorithms for quadratic zero–one programs on the hypercube architecture. Ann. Oper. Res. 22(1), 271–292 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Billionnet, A., Sutter, A.: Minimization of a quadratic pseudo-boolean function. Eur. J. Oper. Res. 78(1), 106–115 (1994)

    Article  MATH  Google Scholar 

  21. Palubeckis, G.: A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming. Computing 54(4), 283–301 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Prog. 82(3), 291–315 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hansen, P., Jaumard, B., Meyer, C., Groupe, Q.: d’études et de recherche en analyse des décisions (Montréal, Exact sequential algorithms for additive clustering. Montréal: Groupe d’études et de recherche en analyse des décisions (2000)

  24. Huang, H.-X., Pardalos, P., Prokopyev, O.: Lower bound improvement and forcing rule for quadratic binary programming. Comput. Optim. Appl. 33(2–3), 187–208 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pardalos, P.M., Prokopyev, O.A., Busygin, S.: Continuous approaches for solving discrete optimization problems. In: Handbook on Modelling for Discrete Optimization, pp. 39–60, Springer (2006)

  26. Pan, S., Tan, T., Jiang, Y.: A global continuation algorithm for solving binary quadratic programming problems. Comput. Optim. Appl. 41(3), 349–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gueye, S., Michelon, P.: A linearization framework for unconstrained quadratic (0–1) problems. Discret. Appl. Math. 157(6), 1255–1266 (2009). Reformulation Techniques and Mathematical Programming

    Article  MathSciNet  MATH  Google Scholar 

  28. Pham Dinh, T., Nguyen Canh, N., Le Thi, H.: An efficient combined dca and bnb using dc/sdp relaxation for globally solving binary quadratic programs. J. Global Optim. 48(4), 595–632 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mauri, G.R., Lorena, L.A.N.: Lagrangean decompositions for the unconstrained binary quadratic programming problem. Int. Trans. Oper. Res. 18(2), 257–270 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mauri, G.R., Lorena, L.A.N.: A column generation approach for the unconstrained binary quadratic programming problem. Eur. J. Oper. Res. 217(1), 69–74 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mauri, G.R., Lorena, L.A.N.: Improving a lagrangian decomposition for the unconstrained binary quadratic programming problem. Comput. Oper. Res. 39(7), 1577–1581 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, D., Sun, X., Liu, C.: An exact solution method for unconstrained quadratic 0–1 programming: a geometric approach. J. Glob. Optim. 52(4), 797–829 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Beasley, J.: Heuristic algorithms for the unconstrained binary quadratic programming problem (1998)

  34. Glover, F., Kochenberger, G.A., Alidaee, B.: Adaptive memory tabu search for binary quadratic programs. Manag. Sci. 44(3), 336–345 (1998)

    Article  MATH  Google Scholar 

  35. Glover, F., Kochenberger, G., Alidaee, B., Amini, M.: Meta-Heuristics. Tabu search with critical event memory: an enhanced application for binary quadratic programs, pp. 93–109. Springer, New York (1999)

    Google Scholar 

  36. Palubeckis, G.: Iterated tabu search for the unconstrained binary quadratic optimization problem. Informatica 17(2), 279–296 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Glover, F., Lü, Z., Hao, J.-K.: Diversification-driven tabu search for unconstrained binary quadratic problems. 4OR 8(3), 239–253 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lü, Z., Hao, J.-K., Glover, F.: Evolutionary Computation in Combinatorial Optimization. A study of memetic search with multi-parent combination for UBQP, pp. 154–165. Springer, Heidelberg (2010)

    Book  Google Scholar 

  39. Shylo, V., Shylo, O.: Systems analysis; solving unconstrained binary quadratic programming problem by global equilibrium search. Cyber. Syst. Anal. 47(6), 889–897 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lü, Z., Hao, J.-K., Glover, F.: Neighborhood analysis: a case study on curriculum-based course timetabling. J. Heuristics 17(2), 97–118 (2011)

    Article  Google Scholar 

  41. Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: Probabilistic grasp-tabu search algorithms for the UBQP problem. Comput. Oper. Res. 40(12), 3100–3107 (2013)

    Article  MathSciNet  Google Scholar 

  42. Alkhamis, T.M., Hasan, M., Ahmed, M.A.: Simulated annealing for the unconstrained quadratic pseudo-boolean function. Eur. J. Oper. Res. 108(3), 641–652 (1998)

    Article  MATH  Google Scholar 

  43. Katayama, K., Narihisa, H.: Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem. Eur. J. Oper. Res. 134(1), 103–119 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Merz, P., Freisleben, B.: Genetic algorithms for binary quadratic programming. In: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, pp. 417–424, Citeseer, (1999)

  45. Katayama, K., Tani, M., Narihisa, H.: Solving large binary quadratic programming problems by effective genetic local search algorithm. In: GECCO, pp. 643–650 (2000)

  46. Lodi, A., Allemand, K., Liebling, T.M.: An evolutionary heuristic for quadratic 0–1 programming. Eur. J. Oper. Res. 119(3), 662–670 (1999)

    Article  MATH  Google Scholar 

  47. Cai, Y., Wang, J., Yin, J., Zhou, Y.: Memetic clonal selection algorithm with \(\{{\rm EDA}\}\) vaccination for unconstrained binary quadratic programming problems. Expert Syst. Appl. 38(6), 7817–7827 (2011)

    Article  Google Scholar 

  48. Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: A multilevel algorithm for large unconstrained binary quadratic optimization. In: Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems, pp. 395–408, Springer (2012)

  49. Amini, M.M., Alidaee, B., Kochenberger, G.A.: New Ideas in Optimization. A scatter search approach to unconstrained quadratic binary programs, pp. 317–330. McGraw-Hill, New York (1999)

    Google Scholar 

  50. Palubeckis, G., Tomkevicius, A.: \(\{{\rm GRASP}\}\) implementations for the unconstrained binary quadratic optimization problem. Inf. Technol. Control 24, 14–20 (2002)

    Google Scholar 

  51. Boros, E., Hammer, P.L., Tavares, G.: Local search heuristics for quadratic unconstrained binary optimization (QUBO). J. Heuristics 13(2), 99–132 (2007)

    Article  Google Scholar 

  52. Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization. Technical Report Rutcor (2006)

  53. Dantzig, G.B., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)

    Article  MATH  Google Scholar 

  54. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  55. Boros, E., Hammer, P.: A max-flow approach to improved roof duality in quadratic 0-1 minimization. Rutgers University. Rutgers Center for Operations Research (RUTCOR) (1989)

  56. Chardaire, P., Sutter, A.: A decomposition method for quadratic zero-one programming. Manag. Sci. 41(4), 704–712 (1995)

    Article  MATH  Google Scholar 

  57. Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization (2006)

  58. Bian, Z., Chudak, F., Israel, R., Lackey, B., Macready, W.G., Roy, A.: Discrete optimization using quantum annealing on sparse Ising models. Interdiscip. Phys. 2, 56 (2014)

    Google Scholar 

  59. Zintchenko, I., Hastings, M.B., Troyer, M.: From local to global ground states in ising spin glasses. Phys. Rev. B 91(2), 24201 (2015)

    Article  Google Scholar 

  60. Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58(5), 5355 (1998)

    Article  Google Scholar 

  61. Finnila, A.B., Gomez, M.A., Sebenik, C., Stenson, C., Doll, J.D.: Quantum annealing: a new method for minimizing multidimensional functions. Chem. Phys. Lett. 219(5), 343–348 (1994)

    Article  Google Scholar 

  62. Ray, P., Chakrabarti, B.K., Chakrabarti, A.: Sherrington-kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations. Phys. Rev. B 39(16), 11828 (1989)

    Article  Google Scholar 

  63. Santoro, G.E., Martoňák, R., Tosatti, E., Car, R.: Theory of quantum annealing of an Ising spin glass. Science 295(5564), 2427–2430 (2002)

    Article  Google Scholar 

  64. Martoňák, R., Santoro, G.E., Tosatti, E.: Quantum annealing by the path-integral monte carlo method: the two-dimensional random ising model. Phys. Rev. B 66, 094203 (2002)

    Article  Google Scholar 

  65. Crosson, E., Harrow, A. W.: Simulated quantum annealing can be exponentially faster than classical simulated annealing, arXiv preprint arXiv:1601.03030 (2016)

  66. Battaglia, D.A., Santoro, G.E., Tosatti, E.: Optimization by quantum annealing: lessons from hard satisfiability problems. Phys. Rev. E 71(6), 066707 (2005)

    Article  Google Scholar 

  67. Lanting, T., Przybysz, A.J., Smirnov, A.Y., Spedalieri, F.M., Amin, M.H., Berkley, A.J., Harris, R., Altomare, F., Boixo, S., Bunyk, P., et al.: Entanglement in a quantum annealing processor. Phys. Rev. X 4(2), 021041 (2014)

    Google Scholar 

  68. Neven, H., Smelyanskiy, V. N., Boixo, S., Shabani, A., Isakov, S. V., Dykman, M., Denchev, V. S., Amin, M., Smirnov, A., Mohseni, M.: Computational role of collective tunneling in a quantum annealer. Bull. Am. Phys. Soc. 60(1) (2015)

  69. McGeoch, C. C., Wang, C.: Experimental evaluation of an adiabiatic quantum system for combinatorial optimization. In: Proceedings of the ACM International Conference on Computing Frontiers, ACM (2013)

  70. Katzgraber, H.G., Hamze, F., Andrist, R.S.: Glassy chimeras could be blind to quantum speedup: designing better benchmarks for quantum annealing machines. Phys. Rev. X 4(2), 021008 (2014)

    Google Scholar 

  71. Boixo, S., Rønnow, T.F., Isakov, S.V., Wang, Z., Wecker, D., Lidar, D.A., Martinis, J.M., Troyer, M.: Evidence for quantum annealing with more than one hundred qubits. Nat. Phys. 10(3), 218–224 (2014)

    Article  Google Scholar 

  72. Hen, I., Job, J., Albash, T., Rønnow, T.F., Troyer, M., Lidar, D.A.: Probing for quantum speedup in spin-glass problems with planted solutions. Phys. Rev. A 92(4), 042325 (2015)

    Article  Google Scholar 

  73. King, A.D.: Performance of a quantum annealer on range-limited constraint satisfaction problems, arXiv preprint. arXiv:1502.02098 (2015)

  74. Denchev, V.S., Boixo, S., Isakov, S.V., Ding, N., Babbush, R., Smelyanskiy, V., Martinis, J., Neven, H.: What is the computational value of finite range tunneling?, arXiv preprint. arXiv:1512.02206 (2015)

  75. Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science 345(6195), 420–424 (2014)

    Article  Google Scholar 

  76. Martin-Mayor, V., Hen, I.: Unraveling quantum annealers using classical hardness. Sci. Rep. 5 (2015)

  77. Katzgraber, H.G., Hamze, F., Zhu, Z., Ochoa, A.J., Munoz-Bauza, H.: Seeking quantum speedup through spin glasses: the good, the bad, and the ugly. Phys. Rev. X 5(3), 031026 (2015)

    Google Scholar 

  78. Bunyk, P., Hoskinson, E.M., Johnson, M., Tolkacheva, E., Altomare, F., Berkley, A.J., Harris, R., Hilton, J.P., Lanting, T., Przybysz, A.J., Whittaker, J.: Architectural considerations in the design of a superconducting quantum annealing processor. IEEE Trans. Appl. Supercond. 24, 1–10 (2014)

    Article  Google Scholar 

  79. Choi, V.: Minor-embedding in adiabatic quantum computation: I. the parameter setting problem. Quant. Inf. Process. 7(5), 193–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  80. Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quant. Inf. Process. 10(3), 343–353 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  81. Williams, C.: State-of-the-art quantum annealing and its application to cryptology, Isaac Newton Institute (2014). http://sms.cam.ac.uk/media/1804114

  82. Merz, P., Katayama, K.: Memetic algorithms for the unconstrained binary quadratic programming problem. BioSystems 78(1), 99–118 (2004)

    Article  Google Scholar 

  83. Glover, F., Hao, J.-K.: Efficient evaluations for solving large 0–1 unconstrained quadratic optimisation problems. Int. J. Metaheuristics 1(1), 3–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: Path relinking for unconstrained binary quadratic programming. Eur. J. Oper. Res. 223(3), 595–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  85. Beasley, J.E.: OR-Library: Unconstrained binary quadratic programming,” (2014). http://people.brunel.ac.uk/~mastjjb/jeb/orlib/bqpinfo.html

  86. Tavares, G.: New algorithms for Quadratic Unconstrained Binary Optimization (QUBO) with applications in engineering and social sciences. PhD thesis, Rutgers University, Graduate School - New Brunswick (2008)

  87. Palubeckis, G.: Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann. Oper. Res. 131(1–4), 259–282 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  88. Pudenz, K.L., Albash, T., Lidar, D.A.: Error-corrected quantum annealing with hundreds of qubits. Nat. Commun. 5 (2014)

  89. Pudenz, K.L., Albash, T., Lidar, D.A.: Quantum annealing correction for random ising problems. Phys. Rev. A 91, 042302 (2015)

    Article  Google Scholar 

  90. Barends, R., Lamata, L., Kelly, J., Garcia-Alvarez, L., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Hoi, I.C., Neill, C., O/’Malley, P.J.J., Quintana, C., Roushan, P., Vainsencher, A., Wenner, J., Solano, E., Martinis, J.M.: Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6, 07 (2015)

  91. Lechner, W., Hauke, P., Zoller, P.: A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1(9), e1500838 (2015)

    Article  Google Scholar 

  92. Pastawski, F., Preskill, J.: Error correction for a proposed quantum annealing architecture, arXiv preprint. arXiv:1511.00004 (2015)

Download references

Acknowledgments

The authors would like to thank Marko Bucyk for editing a draft of this paper and Robyn Foerster, Phil Goddard, and Pooya Ronagh for their useful comments. This work was supported by 1QB Information Technologies (1QBit) and Mitacs.

Author information

Authors and Affiliations

  1. 1QB Information Technologies (1QBit), 458–550 Burrard Street, Vancouver, BC, V6C 2B5, Canada

    Gili Rosenberg, Mohammad Vazifeh & Brad Woods

  2. Department of Mathematics and Earth and Ocean Science, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Eldad Haber

Authors
  1. Gili Rosenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Vazifeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brad Woods
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eldad Haber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gili Rosenberg.

Ethics declarations

Conflicts of interest

EH declares no conflict of interest. GR, BW, and MV were academic interns at 1QBit when the work was done, and GR and BW are still at 1QBit. 1QBit is focused on solving real-world problems using quantum computers. D-Wave Systems is a minority investor in 1QBit.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosenberg, G., Vazifeh, M., Woods, B. et al. Building an iterative heuristic solver for a quantum annealer. Comput Optim Appl 65, 845–869 (2016). https://doi.org/10.1007/s10589-016-9844-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-016-9844-y

Keywords

Search

Navigation