Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study

  • Gabriela Ochoa
  • Nadarajen Veerapen
  • Fabio Daolio
  • Marco Tomassini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10197)

Abstract

Phase transitions play an important role in understanding search difficulty in combinatorial optimisation. However, previous attempts have not revealed a clear link between fitness landscape properties and the phase transition. We explore whether the global landscape structure of the number partitioning problem changes with the phase transition. Using the local optima network model, we analyse a number of instances before, during, and after the phase transition. We compute relevant network and neutrality metrics; and importantly, identify and visualise the funnel structure with an approach (monotonic sequences) inspired by theoretical chemistry. While most metrics remain oblivious to the phase transition, our results reveal that the funnel structure clearly changes. Easy instances feature a single or a small number of dominant funnels leading to global optima; hard instances have a large number of suboptimal funnels attracting the search. Our study brings new insights and tools to the study of phase transitions in combinatorial optimisation.

References

  1. 1.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)MATHGoogle Scholar
  2. 2.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)MATHGoogle Scholar
  3. 3.
    Gent, I.P., Walsh, T.: The SAT phase transition. In: Proceedings of ECAI 1996, vol. 94, pp. 105–109. PITMAN (1994)Google Scholar
  4. 4.
    Culberson, J., Gent, I.P.: Frozen development in graph coloring. Theor. Comput. Sci. 265(1), 227–264 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gent, I.P., Walsh, T.: Phase transitions and annealed theories: number partitioning as a case study. In: Proceedings of ECAI 1996, pp. 170–174. PITMAN (1996)Google Scholar
  6. 6.
    Gomes, C., Walsh, T.: Randomness and structure. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, vol. 2, pp. 639–664. Elsevier, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Kambhampati, S.C., Liu, T.: Phase transition and network structure in realistic SAT problems. In: Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence, AAAI 2013, pp. 1619–1620. AAAI Press (2013)Google Scholar
  8. 8.
    Martin, O.C., Monasson, R., Zecchina, R.: Statistical mechanics methods and phase transitions in optimization problems. Theor. Comput. Sci. 265(1), 3–67 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tomassini, M., Vérel, S., Ochoa, G.: Complex-network analysis of combinatorial spaces: the NK landscape case. Phys. Rev. E 78(6), 066114 (2008)CrossRefGoogle Scholar
  10. 10.
    Verel, S., Ochoa, G., Tomassini, M.: Local optima networks of NK landscapes with neutrality. IEEE Trans. Evol. Comput. 15(6), 783–797 (2011)CrossRefGoogle Scholar
  11. 11.
    Doye, J.P.K., Miller, M.A., Wales, D.J.: The double-funnel energy landscape of the 38-atom Lennard-Jones cluster. J. Chem. Phys. 110(14), 6896–6906 (1999)CrossRefGoogle Scholar
  12. 12.
    Lunacek, M., Whitley, D., Sutton, A.: The impact of global structure on search. In: Rudolph, G., Jansen, T., Beume, N., Lucas, S., Poloni, C. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 498–507. Springer, Heidelberg (2008). doi:10.1007/978-3-540-87700-4_50 CrossRefGoogle Scholar
  13. 13.
    Kerschke, P., Preuss, M., Wessing, S., Trautmann, H.: Detecting funnel structures by means of exploratory landscape analysis. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, GECCO 2015, pp. 265–272. ACM, New York (2015)Google Scholar
  14. 14.
    Ochoa, G., Veerapen, N.: Deconstructing the big valley search space hypothesis. In: Chicano, F., Hu, B., García-Sánchez, P. (eds.) EvoCOP 2016. LNCS, vol. 9595, pp. 58–73. Springer, Heidelberg (2016). doi:10.1007/978-3-319-30698-8_5 CrossRefGoogle Scholar
  15. 15.
    Ochoa, G., Veerapen, N.: Additional dimensions to the study of funnels in combinatorial landscapes. In: Proceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO 2016, pp. 373–380. ACM, New York (2016)Google Scholar
  16. 16.
    Herrmann, S., Ochoa, G., Rothlauf, F.: Communities of local optima as funnels in fitness landscapes. In: Proceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO 2016, pp. 325–331. ACM, New York (2016)Google Scholar
  17. 17.
    Ferreira, F.F., Fontanari, J.F.: Probabilistic analysis of the number partitioning problem. J. Phys. A: Math. Gen. 31(15), 3417 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mertens, S.: Phase transition in the number partitioning problem. Phys. Rev. Lett. 81(20), 4281–4284 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Stadler, P.F., Hordijk, W., Fontanari, J.F.: Phase transition and landscape statistics of the number partitioning problem. Phys. Rev. E 67(5), 056701 (2003)CrossRefGoogle Scholar
  20. 20.
    Flamm, C., Hofacker, I.L., Stadler, P.F., Wolfinger, M.T.: Barrier trees of degenerate landscapes. Z. Phys. Chem. (Int. J. Res. Phys. Chem. Chem. Phy.) 216(2/2002), 155–173 (2002)Google Scholar
  21. 21.
    Alyahya, K., Rowe, J.E.: Phase transition and landscape properties of the number partitioning problem. In: Blum, C., Ochoa, G. (eds.) EvoCOP 2014. LNCS, vol. 8600, pp. 206–217. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44320-0_18 Google Scholar
  22. 22.
    Boese, K.D., Kahng, A.B., Muddu, S.: A new adaptive multi-start technique for combinatorial global optimizations. Oper. Res. Lett. 16(2), 101–113 (1994)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hains, D.R., Whitley, L.D., Howe, A.E.: Revisiting the big valley search space structure in the TSP. J. Oper. Res. Soc. 62(2), 305–312 (2011)CrossRefGoogle Scholar
  24. 24.
    Berry, R.S., Kunz, R.E.: Topography and dynamics of multidimensional interatomic potential surfaces. Phys. Rev. Lett. 74, 3951–3954 (1995)CrossRefGoogle Scholar
  25. 25.
    Wales, D.J.: Energy landscapes and properties of biomolecules. Phys. Biol. 2(4), S86–S93 (2005)CrossRefGoogle Scholar
  26. 26.
    Becker, O.M., Karplus, M.: The topology of multidimensional potential energy surfaces: theory and application to peptide structure and kinetics. J. Chem. Phys. 106(4), 1495 (1997)CrossRefGoogle Scholar
  27. 27.
    Mézard, M., Mora, T., Zecchina, R.: Clustering of solutions in the random satisfiability problem. Phys. Rev. Lett. 94, 197205 (2005)CrossRefGoogle Scholar
  28. 28.
    Stadler, P.F.: Fitness landscapes. Appl. Math. Comput. 117, 187–207 (2002)MathSciNetGoogle Scholar
  29. 29.
    Huynen, M.A., Stadler, P.F., Fontana, W.: Smoothness within ruggedness: the role of neutrality in adaptation. Proc. Nat. Acad. Sci. U.S.A. 93(1), 397–401 (1996)CrossRefGoogle Scholar
  30. 30.
    Barnett, L.: Ruggedness and neutrality - the NKp family of fitness landscapes. In: Adami, C., Belew, R.K., Kitano, H., Taylor, C. (eds.) Proceedings of the Sixth International Conference on Artificial Life, ALIFE VI, pp. 18–27. The MIT Press, Cambridge (1998)Google Scholar
  31. 31.
    Daolio, F., Verel, S., Ochoa, G., Tomassini, M.: Local optima networks of the quadratic assignment problem. In: 2010 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8 (2010)Google Scholar
  32. 32.
    Daolio, F., Tomassini, M., Vérel, S., Ochoa, G.: Communities of minima in local optima networks of combinatorial spaces. Phys. A: Stat. Mech. Appl. 390(9), 1684–1694 (2011)CrossRefGoogle Scholar
  33. 33.
    Auger, A., Hansen, N.: Performance evaluation of an advanced local search evolutionary algorithm. In: The 2005 IEEE Congress on Evolutionary Computation, vol. 2, pp. 1777–1784. IEEE (2005)Google Scholar
  34. 34.
    Mertens, S.: The easiest hard problem: number partitioning. In: Percus, A., Istrate, G., Moore, C. (eds.) Computational Complexity and Statistical Physics. The Santa Fe Institute Studies in the Sciences of Complexity, vol. 125, pp. 125–139. Oxford University Press, New York (2006)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Gabriela Ochoa
    • 1
  • Nadarajen Veerapen
    • 1
  • Fabio Daolio
    • 1
  • Marco Tomassini
    • 2
  1. 1.Computing Science and MathematicsUniversity of StirlingStirlingUK
  2. 2.Faculty of Business and Economics, Information Systems DepartmentUniversity of LausanneLausanneSwitzerland

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