Journal of Heuristics

, Volume 19, Issue 4, pp 711–728 | Cite as

Elementary landscape decomposition of the 0-1 unconstrained quadratic optimization



Landscapes’ theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of an especial kind of landscape called elementary landscape. The elementary landscape decomposition of a combinatorial optimization problem is a useful tool for understanding the problem. Such decomposition provides an additional knowledge on the problem that can be exploited to explain the behavior of some existing algorithms when they are applied to the problem or to create new search methods for the problem. In this paper we analyze the 0-1 Unconstrained Quadratic Optimization from the point of view of landscapes’ theory. We prove that the problem can be written as the sum of two elementary components and we give the exact expressions for these components. We use the landscape decomposition to compute autocorrelation measures of the problem, and show some practical applications of the decomposition.


Fitness landscapes Unconstrained quadratic optimization Landscapes theory Autocorrelation length 


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  1. Alidaee, B., Kochenberger, G., Ahmadian, A.: 0-1 quadratic programming approach for the optimal solution of two scheduling problems. Int. J. Syst. Sci. 25, 401–408 (1994) MathSciNetMATHCrossRefGoogle Scholar
  2. Amini, M.M., Alidaee, B., Kochenberger, G. New Ideas in Optimisation. McGraw-Hill, London (1999). Chap. A Scatter Search Approach to Unconstrained Quadratic Binary Programs, pp 317–329 Google Scholar
  3. Angel, E., Zissimopoulos, V.: Autocorrelation coefficient for the graph bipartitioning problem. Theor. Comput. Sci. 191, 229–243 (1998) MathSciNetMATHCrossRefGoogle Scholar
  4. Angel, E., Zissimopoulos, V.: On the classification of NP-complete problems in terms of their correlation coefficient. Discrete Appl. Math. 99, 261–277 (2000a) MathSciNetMATHCrossRefGoogle Scholar
  5. Angel, E., Zissimopoulos, V.: On the landscape ruggedness of the quadratic assignment problem. Theor. Comput. Sci. 263, 159–172 (2000b) MathSciNetCrossRefGoogle Scholar
  6. Biyikoglu, T., Leyold, J., Stadler, P.F.: Laplacian Eigenvectors of Graphs. Lecture Notes in Mathematics. Springer, Berlin (2007) MATHCrossRefGoogle Scholar
  7. Chartaire, P., Sutter, A.: A decomposition method for quadratic 0-1 minimization. Manag. Sci. 41(4), 704–712 (1994) CrossRefGoogle Scholar
  8. Codenotti, B., Margara, L.: Local properties of some NP-complete problems. Tech. Rep. TR 92-021, International Computer Science Institute, Berkeley, USA (1992) Google Scholar
  9. Gallo, G., Hammer, P., Simeone, B.: Quadratic knapsack problems. Math. Program. 12, 132–149 (1980) MathSciNetMATHGoogle Scholar
  10. García-Pelayo, R., Stadler, P.: Correlation length, isotropy and meta-stable states. Physica D 107(2–4), 240–254 (1997) CrossRefGoogle Scholar
  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979) MATHGoogle Scholar
  12. Glover, F., Alidaee, B., Rego, C., Kochenberger, G.: One-pass heuristics for large-scale unconstrained binary quadratic problems. Eur. J. Oper. Res. 137(2), 272–287 (2002) MathSciNetMATHCrossRefGoogle Scholar
  13. Grover, L.K.: Local search and the local structure of NP-complete problems. Oper. Res. Lett. 12, 235–243 (1992) MathSciNetMATHCrossRefGoogle Scholar
  14. Harary, F.: On the notion of balanced of a signed graphs. Mich. Math. J. 2(2), 143–146 (1953) CrossRefGoogle Scholar
  15. Heckendorn, R.B., Rana, S.B., Whitley, L.D.: Test function generators as embedded landscapes. In: Banzhaf, W., Reeves, C.R. (eds.) FOGA, pp. 183–198. Morgan Kaufmann, San Mateo (1998) Google Scholar
  16. Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998) MathSciNetMATHGoogle Scholar
  17. Katayama, K., Narihisa, H.: Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem. Eur. J. Oper. Res. 134, 103–119 (2001) MathSciNetMATHCrossRefGoogle Scholar
  18. Krarup, J., Pruzan, A.: Computer aided layout design. Math. Program. Stud. 9, 75–94 (1978) MathSciNetCrossRefGoogle Scholar
  19. Liu, W., Wilkins, D., Alidaee, B.: A hybrid multi-exchange local search for unconstrained binary quadratic program. Tech. Rep. HCES-09-05, Hearin Center for Enterprise Science, University of Mississippi (2005) Google Scholar
  20. Lu, G., Bahsoon, R., Yao, X.: Applying elementary landscape analysis to search-based software engineering. In: Proceedings of the 2nd International Symposium on Search Based Software Engineering (SSBSE) (2010) Google Scholar
  21. McBride, R.D., Yormack, J.S.: An implicit enumeration algorithm for quadratic integer programming. Manag. Sci. 26, 282–296 (1980) MATHCrossRefGoogle Scholar
  22. Merz, P., Freisleben, B.: Greedy and local search heuristics for unconstrained binary quadratic programming. J. Heuristics 8(2), 197–213 (2002) MATHCrossRefGoogle Scholar
  23. Merz, P., Katayama, K.: Memetic algorithms for the unconstrained binary quadratic programming problem. Biosystems 78(1–3), 99–118 (2004) CrossRefGoogle Scholar
  24. Palubeckis, G.: Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann. Oper. Res. 131, 259–282 (2004) MathSciNetMATHCrossRefGoogle Scholar
  25. Palubeckis, G., Tomkevièius, A.: Grasp implementations for the uncostrained binary quadratic optimization problem. Inf. Technol. Control 24(3), 14–20 (2002) Google Scholar
  26. Philips, A.T., Rosen, J.B.: A quadratic assignment formulation of the molecular conformation problem. J. Glob. Optim. 4, 229–241 (1994) CrossRefGoogle Scholar
  27. Stadler, P.F.: Toward a theory of landscapes. In: López-Peña, R., Capovilla, R., García-Pelayo, R., Waelbroeck, H., Zertruche, F. (eds.) Complex Systems and Binary Networks, pp. 77–163. Springer, Berlin (1995) Google Scholar
  28. Stadler, P.F.: Landscapes and their correlation functions. J. Math. Chem. 20, 1–45 (1996) MathSciNetMATHCrossRefGoogle Scholar
  29. Stadler, P.F.: Biological Evolution and Statistical Physics. Springer, Berlin (2002). Chap. Fitness Landscapes, pp. 183–204 Google Scholar
  30. Sutton, A.M., Whitley, L.D., Howe, A.E.: A polynomial time computation of the exact correlation structure of k-satisfiability landscapes. In: Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation, pp. 365–372. ACM, New York (2009) Google Scholar
  31. Sutton, A.M., Howe, A.E., Whitley, L.D.: Directed plateau search for MAX-k-SAT. In: Proceedings of Symposium on Combinatorial Search, Atlanta, GA, USA (2010) Google Scholar
  32. Sutton, A.M., Whitley, L.D., Howe, A.E.: Computing the moments of k-bounded pseudo-boolean functions over hamming spheres of arbitrary radius in polynomial time. Theor. Comput. Sci.. doi: 10.1016/j.tcs.2011.02.006 (2011) Google Scholar
  33. Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45(1), 5–24 (1923) MATHCrossRefGoogle Scholar
  34. Weinberger, E.: Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern. 63(5), 325–336 (1990) MATHCrossRefGoogle Scholar
  35. Whitley, D., Sutton, A.M., Howe, A.E.: Understanding elementary landscapes. In: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, pp. 585–592. ACM, New York (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.E.T.S. Ingeniería InformáticaUniversity of MálagaMálagaSpain

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