A Complex-Networks View of Hard Combinatorial Search Spaces

  • Marco TomassiniEmail author
  • Fabio Daolio
Part of the Studies in Computational Intelligence book series (SCI, volume 447)


According to worst-case complexity analysis, difficult combinatorial problems are those for which no polynomial-time algorithms are known (see, for instance, [15]). Thus, according to this point of view, large enough instances of these problems cannot be solved in reasonable time. The mathematical analysis is primarily based on decision problems, i.e. those that require a yes/no answer [7, 15], but the theory can readily be extended to optimization problems [16], roughly speaking, those in which we seek a solution with an associated minimum or maximum cost, which are the ones that will be dealt with here.


Local Search Local Optimum Travel Salesman Problem Community Detection Fitness Landscape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A.: Characterization and modeling of weighted networks. Physica A 346, 34–43 (2005)CrossRefGoogle Scholar
  2. 2.
    Clauset, A., Newman, M.E.J., Moore, C.: Finding community structure in very large networks. Physical Review E 70(6), 66111 (2004)CrossRefGoogle Scholar
  3. 3.
    Daolio, F., Tomassini, M., Vérel, S., Ochoa, G.: Communities of minima in local optima networks of combinatorial spaces. Physica A 390, 1684–1694 (2011)CrossRefGoogle Scholar
  4. 4.
    Daolio, F., Vérel, S., Ochoa, G., Tomassini, M.: Local optima networks of the quadratic assignment problem. In: IEEE Congress on Evolutionary Computation, CEC 2010, pp. 3145–3152. IEEE Press (2010)Google Scholar
  5. 5.
    Derrida, B.: Random energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24, 2613 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fortunato, S.: Community detection in graphs. Physics Reports 486, 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  8. 8.
    Garnier, J., Kallel, L.: Efficiency of local search with multiple local optima. SIAM Journal on Discrete Mathematics 15(1), 122–141 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jones, T.: Evolutionary algorithms, fitness landscapes and search. PhD thesis. The University of New Mexico (1995)Google Scholar
  10. 10.
    Kallel, L., Naudts, B., Rogers, A. (eds.): Theoretical Aspects of Evolutionary Computing. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  11. 11.
    Kauffman, S.A.: The Origins of Order. Oxford University Press, New York (1993)Google Scholar
  12. 12.
    Knowles, J.D., Corne, D.W.: Instance Generators and Test Suites for the Multiobjective Quadratic Assignment Problem. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 295–310. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  14. 14.
    Ochoa, G., Tomassini, M., Vérel, S., Darabos, C.: A study of NK landscapes’ basins and local optima networks. In: Genetic and Evolutionary Computation Conference, GECCO 2008, pp. 555–562. ACM (2008)Google Scholar
  15. 15.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  16. 16.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)zbMATHGoogle Scholar
  17. 17.
    Reichardt, J., Bornholdt, S.: Statistical mechanics of community detection. Physical Review E 74(1), 16110 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Reidys, C.M., Stadler, P.F.: Combinatorial landscapes. SIAM Review 44(1), 3–54 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Taillard, E.D.: Comparison of iterative searches for the quadratic assignment problem. Location Science 3(2), 87–105 (1995)zbMATHCrossRefGoogle Scholar
  20. 20.
    Talbi, E.-G.: Metaheuristics: From Design to Implementation. Wiley, Hoboken (2009)zbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    Vérel, S., Ochoa, G., Tomassini, M.: Local optima networks of NK landscapes with neutrality. IEEE Trans. Evol. Comp. 15(6), 783–797 (2011)CrossRefGoogle Scholar
  23. 23.
    Weinberger, E.D.: Correlated and uncorrelated fitness landscapes and how to tell the difference. Biological Cybernetics 63, 325–336 (1990)zbMATHCrossRefGoogle Scholar

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© Springer Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Business and Economics, Information System DepartmentUniversity of LausanneLausanneSwitzerland

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