Journal of Combinatorial Optimization

, Volume 38, Issue 4, pp 1213–1262 | Cite as

Random walk’s correlation function for multi-objective NK landscapes and quadratic assignment problem

  • Madalina M. DruganEmail author


The random walk’ correlation matrix of multi-objective combinatorial optimization problems utilizes both local structure and general statistics of the objective functions. Reckoning time of correlation, or the random walk of lag 0, is quadratic in problem size L and number of objectives D. The computational complexity of the correlation coefficients of mNK is \(O(D^2 K^2 L)\), and of mQAP is \(O(D^2 L^2)\), where K is the number of interacting bits. To compute the random walk of a lag larger than 0, we employ a weighted graph Laplacian that associates a mutation operator with the difference in the objective function. We calculate the expected objective vector of a neighbourhood function and the eigenvalues of the corresponding transition matrix. The computational complexity of random walk’s correlation coefficients is polynomial with the problem size L and the number of objectives D. The computational effort of the random walks correlation coefficients of mNK is \(O(2^K L D^2)\), whereas of mQAP is \(O(L^6 D^2)\). Numerical examples demonstrate the utilization of these techniques.


Random walk Multi-objective optimization Neighbourhood function Graph Laplacian Matrix theory 



  1. Aguirre HE, Tanaka K (2007) Working principles, behavior, and performance of MOEAs on MNK-landscapes. Eur J Oper Res 181(3):1670–1690zbMATHCrossRefGoogle Scholar
  2. Aleti A, Moser I, Grunske L (2017) Analysing the fitness landscape of search-based software testing problems. Autom Softw Eng 24(3):603–621CrossRefGoogle Scholar
  3. Alyahya K, Rowe JE (2019) Landscape analysis of a class of NP-hard binary packing problems. Evol Comput 27(1):47–73CrossRefGoogle Scholar
  4. Angel E, Zissimopoulos V (2002) On the hardness of the quadratic assignment problem with metaheuristics. J Heuristics 8(4):399–414CrossRefGoogle Scholar
  5. Barnes JW, Dimova B, Dokov SP, Solomon A (2003) The theory of elementary landscapes. Appl Math Lett 16(3):337–343MathSciNetzbMATHCrossRefGoogle Scholar
  6. Basseur M, Goeffon A (2015) Climbing combinatorial fitness landscapes. Appl Soft Comput 30:688–704CrossRefGoogle Scholar
  7. Biyikoglu T, Leydold J, Stadler PF (2007) Laplacian eigenvectors of graphs: Perron–Frobenius and Faber–Krahn type theorems. Springer, New YorkzbMATHCrossRefGoogle Scholar
  8. Blot A, Hoos HH, Kessaci ME, Jourdan L (2018) Automatic configuration of bi-objective optimisation algorithms: impact of correlation between objectives. In: International conference on tools with artificial intelligence (ICTAI). IEEE, pp 571–578Google Scholar
  9. Cela E (1997) The quadratic assignment problem. Springer, DordrechtzbMATHGoogle Scholar
  10. Cheng R, Li M, Li K, Yao X (2017) Evolutionary multiobjective optimization based multimodal optimization: fitness landscape approximation and peak detection. Trans Evol Comput. IEEEGoogle Scholar
  11. Chicano F, Whitley LD, Alba E (2011) A methodology to find the elementary landscape decomposition of combinatorial optimization problems. Evol Comput 19(4):597–637CrossRefGoogle Scholar
  12. Chicano F, Luque G, Alba E (2012) Autocorrelation measures for the quadratic assignment problem. Appl Math Lett 25(4):698–705MathSciNetzbMATHCrossRefGoogle Scholar
  13. Chung FRK (1994) Spectral graph theory, vol. 92. CBMSGoogle Scholar
  14. Daolio F, Liefooghe A, Verel S, Aguirre HE, Tanaka K (2015) Global vs local search on multi-objective NK-landscapes: contrasting the impact of problem features. In: Conference on genetic and evolutionary computation (GECCO). ACM, pp 369–376Google Scholar
  15. Daolio F, Liefooghe A, Verel S, Aguirre HE, Tanaka K (2017) Problem features versus algorithm performance on rugged multiobjective combinatorial fitness landscapes. Evol Comput 25(4). MITCrossRefGoogle Scholar
  16. Das KC (2004) The Laplacian spectrum of a graph. Comput Math Appl 48:715–724MathSciNetzbMATHCrossRefGoogle Scholar
  17. Draskoczy B (2010) Fitness distance correlation and search space analysis for permutation based problems. Evolutionary Computation in Combinatorial Optimization EvoCOP, pp 47–58. SpringerGoogle Scholar
  18. Ehrgott M (2005) Multicriteria optimization. Springer, BerlinzbMATHGoogle Scholar
  19. Fontana W, Stadler PF, Bornberg-Bauer EG, Griesmacher T, Hofacker IL, Tacker M, Tarazona P, Weinberger ED, Schuster P (1993) RNA folding and combinatory landscapes. Phys Rev E 47(3):20–83CrossRefGoogle Scholar
  20. Garrett JD (2008) Multiobjective fitness landscape analysis and the design of effective memetic algorithms. Ph.D. thesis, University of MemphisGoogle Scholar
  21. Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput OR 13(5):533–549MathSciNetzbMATHCrossRefGoogle Scholar
  22. Happel R, Stadler PF (1996) Canonical approximation of fitness landscapes. Complexity 2(1):53–58CrossRefGoogle Scholar
  23. Herrmann S, Ochoa G, Rothlauf F (2016) Coarse-grained barrier trees of fitness landscapes. Parallel problem solving from nature—PPSN XIV, pp 901–910Google Scholar
  24. Horn RA, Johnson CR (2013) Matrix analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  25. Kauffman SA (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
  26. Kauffman S, Weinberger E (1989) The NK model of rugged fitness landscapes and its application to the maturation of the immune response. J Theor Biol 141(2):211–245CrossRefGoogle Scholar
  27. Knowles JD, Corne D (2003) Instance generators and test suites for the multiobjective quadratic assignment problem. Evolutionary multi-criterion optimization (EMO), pp 295–310Google Scholar
  28. Koopmans T, Beckmann M (1957) Assignment problems and the location of economic activities. Econometrica 25(1):53–76MathSciNetzbMATHCrossRefGoogle Scholar
  29. Lankaites Pinheiro R, Landa-Silva D, Atkin J (2017) A technique based on trade-off maps to visualise and analyse relationships between objectives in optimisation problems. J Multi-Criteria Dec Anal 24(1–2):37–56CrossRefGoogle Scholar
  30. Li R, Emmerich MT, Eggermont J, Back T, Schutz M, Dijkstra J, Reiber JH (2013a) Mixed integer evolution strategies for parameter optimization. Evol Comput 21(1):29–64 MITCrossRefGoogle Scholar
  31. Li J, Guo J-M, Shiu WC (2013b) On the second largest Laplacian eigenvalues of graphs. Linear Algebra Appl 438:2438–2446 ElsevierMathSciNetzbMATHCrossRefGoogle Scholar
  32. Liefooghe A, Derbel B, Verel S, Aguirre H, Tanaka K (2017) A fitness landscape analysis of Pareto local search on bi-objective permutation flowshop scheduling problems. Evolutionary multi-criterion optimization (EMO). SpringerGoogle Scholar
  33. Merz P (2004) Advanced fitness landscape analysis and the performance of memetic algorithms. Evol Comput 12(3):303–325 MITCrossRefGoogle Scholar
  34. Mohar B (1991) The Laplacian spectrum of graphs. Graph theory, combinatorics, and applications, pp 871–898. WileyGoogle Scholar
  35. Moser I, Gheorghita M, Aleti A (2017) Identifying features of fitness landscapes and relating them to problem difficulty. Evol Comput 25(3):407–437 MITCrossRefGoogle Scholar
  36. Pelikan M, Sastry K, Goldberg DE, Butz MV, Hauschild M (2009) Performance of evolutionary algorithms on NK landscapes with nearest neighbor interactions and tunable overlap. In: Conference on genetic and evolutionary computation GECCO. ACM, pp 851–858Google Scholar
  37. Pitzer E, Beham A, Affenzeller M (2012) Generic hardness estimation using fitness and parameter landscapes applied to robust taboo search and the quadratic assignment problem. In: Conference on genetic and evolutionary computation GECCO. ACM, pp 393–400Google Scholar
  38. Reidys CM, Stadler PF (2002) Combinatorial landscapes. SIAM Rev 44(1):3–54MathSciNetzbMATHCrossRefGoogle Scholar
  39. Smith-Miles K, Lopes L (2012) Measuring instance difficulty for combinatorial optimization problems. Comput OR 39(5):875–889MathSciNetzbMATHCrossRefGoogle Scholar
  40. Stadler PF (1996) Landscapes and their correlation functions. J Math Chem 20(1):1–45MathSciNetzbMATHCrossRefGoogle Scholar
  41. Sutton AM, Whitley LD, Howe AE (2012) Computing the moments of k-bounded pseudo-boolean functions over hamming spheres of arbitrary radius in polynomial time. Theor Comput Sci 425:58–74 ElsevierMathSciNetzbMATHCrossRefGoogle Scholar
  42. Tayarani-N MH, Prugel-Bennett A (2015) Quadratic assignment problem: a landscape analysis. Evol Intel 8(4):165–184 SpringerCrossRefGoogle Scholar
  43. Thierens D (2010) the linkage tree genetic algorithm. Parallel Problem solving from nature—PPSN XI. Springer, pp 264–273Google Scholar
  44. van Remortel P, Ceuppens J, Defaweux A, Lenaerts T, Manderick B (2003) Developmental effects on tuneable fitness landscapes. Evolvable systems: from biology to hardware. Springer, pp 117–128Google Scholar
  45. Verel S, Collard P, Clergue M (2003) Where are bottlenecks in NK fitness landscapes? In: The congress on evolutionary computation, (CEC’03). IEEE, pp 273–280Google Scholar
  46. Verel S, Liefooghe A, Jourdan L, Dhaenens C (2013) On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives. Eur J Oper Res (EJOR) 227(2):331–342 ElsevierMathSciNetCrossRefGoogle Scholar
  47. Verel S, Daolio F, Ochoa G, Tomassini M (2018) Sampling local optima networks of large combinatorial search spaces: the QAP case. Parallel problem solving from nature—PPSN XV. Springer, pp 257–268Google Scholar
  48. Weinberger ED (1996) NP completeness of Kauffman?s NK model, a tuneably rugged fitness landscape. Santa Fe Institute Technical ReportsGoogle Scholar
  49. Whitley D, Sutton AM, Ochoa G, Chicano F (2014) The component model for elementary landscapes and partial neighborhoods. Theor Comput Sci 545:59–75MathSciNetzbMATHCrossRefGoogle Scholar
  50. Wilks SS (1947) Mathematical statistics. Princeton University Press, PrincetonzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.ITLearns.OnlineUtrechtThe Netherlands

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