Autocorrelation measures for the quadratic assignment problem

Abstract In this article we provide an exact expression for computing the autocorrelation coefficient ξ and the autocorrelation length l of any arbitrary instance of the Quadratic Assignment Problem (QAP) in polynomial time using its elementary landscape decomposition. We also provide empirical evidence of the autocorrelation length conjecture in QAP and compute the parameters ξ and l for the 137 instances of the QAPLIB. Our goal is to better characterize the difficulty of this important class of problems to ease the future definition of new optimization methods. Also, the advance that this represents helps to consolidate QAP as an interesting and now better understood problem.

[1]  Enrique Alba,et al.  A Methodology to Find the Elementary Landscape Decomposition of Combinatorial Optimization Problems , 2011, Evolutionary Computation.

[2]  Peter F. Stadler,et al.  Correlation length, isotropy and meta-stable states , 1997 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Enrique Alba,et al.  Elementary Components of the Quadratic Assignment Problem , 2011, ArXiv.

[5]  E. Weinberger,et al.  Correlated and uncorrelated fitness landscapes and how to tell the difference , 1990, Biological Cybernetics.

[6]  B. Drossel.  Biological evolution and statistical physics , 2001, cond-mat/0101409.

[7]  Eric Angela,et al.  On the landscape ruggedness of the quadratic assignment problem , 2001 .

[8]  P. Stadler.  Landscapes and their correlation functions , 1996 .

[9]  J. Wesley Barnes,et al.  The theory of elementary landscapes , 2003, Appl. Math. Lett..

[10]  Vassilis Zissimopoulos,et al.  Autocorrelation Coefficient for the Graph Bipartitioning Problem , 1998, Theor. Comput. Sci..

[11]  Enrique Alba,et al.  Elementary landscape decomposition of the quadratic assignment problem , 2010, GECCO '10.

[12]  Vassilis Zissimopoulos,et al.  On the Classification of NP-complete Problems in Terms of Their Correlation Coefficient , 2000, Discret. Appl. Math..

[13]  Vassilis Zissimopoulos,et al.  On the landscape ruggedness of the quadratic assignment problem , 2001, Theor. Comput. Sci..

[14]  Franz Rendl,et al.  QAPLIB – A Quadratic Assignment Problem Library , 1997, J. Glob. Optim..

[15]  Lov K. Grover.  Local search and the local structure of NP-complete problems , 1992, Oper. Res. Lett..