Computing the moments of k-bounded pseudo-Boolean functions over Hamming spheres of arbitrary radius in polynomial time

We show that given a k-bounded pseudo-Boolean function f, one can always compute the cth moment of f over regions of arbitrary radius in Hamming space in polynomial time using algebraic information from the adjacency structure (where k and c are constants). This result has implications for evolutionary algorithms and local search algorithms because information about promising regions of the search space can be efficiently retrieved, even if the cardinality of the region is exponential in the problem size. Finally, we use our results to introduce a method of efficiently calculating the expected fitness of mutations for evolutionary algorithms.

[1]  Stuart A. Kauffman,et al.  ORIGINS OF ORDER , 2019, Origins of Order.

[2]  James P. Crutchfield,et al.  Evolutionary dynamics : exploring the interplay of selection, accident, neutrality, and function , 2003 .

[3]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[4]  Bart Selman,et al.  Local search strategies for satisfiability testing , 1993, Cliques, Coloring, and Satisfiability.

[5]  Andrew M. Sutton,et al.  A polynomial time computation of the exact correlation structure of k-satisfiability landscapes , 2009, GECCO '09.

[6]  Alden H. Wright,et al.  Efficient Linkage Discovery by Limited Probing , 2003, Evolutionary Computation.

[7]  H. Waelbroeck,et al.  Complex Systems and Binary Networks , 1995 .

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

[10]  P. Stadler.  Spectral Landscape Theory , 1999 .

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

[12]  L. Darrell Whitley,et al.  Polynomial Time Summary Statistics for a Generalization of MAXSAT , 1999, GECCO.

[13]  Weinberger,et al.  RNA folding and combinatory landscapes. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Peter Merz,et al.  Advanced Fitness Landscape Analysis and the Performance of Memetic Algorithms , 2004, Evolutionary Computation.

[15]  J. Walsh.  A Closed Set of Normal Orthogonal Functions , 1923 .

[16]  L. Darrell Whitley,et al.  Test Function Generators as Embedded Landscapes , 1998, FOGA.

[17]  Kenneth A. De Jong,et al.  Are Genetic Algorithms Function Optimizers? , 1992, PPSN.

[18]  Christian M. Reidys,et al.  Combinatorial Landscapes , 2002, SIAM Rev..

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

[20]  Kyomin Jung,et al.  Almost Tight Upper Bound for Finding Fourier Coefficients of Bounded Pseudo- Boolean Functions , 2008, COLT.

[21]  Tim Jones.  Evolutionary Algorithms, Fitness Landscapes and Search , 1995 .

[22]  Robert B. Heckendorn.  Embedded Landscapes , 2002, Evolutionary Computation.

[23]  Thomas Stützle,et al.  Stochastic Local Search: Foundations & Applications , 2004 .

[24]  Hillol Kargupta,et al.  Gene Expression and Fast Construction of Distributed Evolutionary Representation , 2001, Evolutionary Computation.

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

[26]  Terry Jones,et al.  Fitness Distance Correlation as a Measure of Problem Difficulty for Genetic Algorithms , 1995, ICGA.