Fitness Function Distributions over Generalized Search Neighborhoods in the q-ary Hypercube

The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.

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

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

[3]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[4]  Andrew M. Sutton,et al.  Approximating the distribution of fitness over hamming regions , 2011, FOGA '11.

[5]  A. Terras.  Fourier Analysis on Finite Groups and Applications: Index , 1999 .

[6]  Thomas Augustin,et al.  Foundations of Probability , 2011, International Encyclopedia of Statistical Science.

[7]  Frank Neumann,et al.  Optimal Fixed and Adaptive Mutation Rates for the LeadingOnes Problem , 2010, PPSN.

[8]  Enrique Alba,et al.  Exact computation of the expectation curves of the bit-flip mutation using landscapes theory , 2011, GECCO '11.

[9]  Robert B. Heckendorn,et al.  Walsh analysis, epistasis, and optimization problem difficulty for evolutionary algorithms , 1999 .

[10]  Enrique Alba,et al.  Exact computation of the expectation surfaces for uniform crossover along with bit-flip mutation , 2014, Theor. Comput. Sci..

[11]  Thomas Bäck,et al.  The Interaction of Mutation Rate, Selection, and Self-Adaptation Within a Genetic Algorithm , 1992, PPSN.

[12]  Thomas Jansen,et al.  Mutation Rate Matters Even When Optimizing Monotonic Functions , 2013, Evolutionary Computation.

[13]  Dennis Stanton,et al.  Zeros of generalized Krawtchouk polynomials , 1990 .

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

[15]  A. Eisenblätter.  Frequency Assignment in GSM Networks: Models, Heuristics, and Lower Bounds , 2001 .

[16]  Carsten Witt,et al.  Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation , 2012, STACS.

[17]  L. Darrell Whitley,et al.  A Tractable Walsh Analysis of SAT and its Implications for Genetic Algorithms , 1998, AAAI/IAAI.

[18]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods On the Choice of the Mutation Probability for the ( 1 + 1 ) EA , 2006 .

[19]  Andrew M. Sutton,et al.  Computing the moments of k-bounded pseudo-Boolean functions over Hamming spheres of arbitrary radius in polynomial time , 2012, Theor. Comput. Sci..

[20]  Andrew M. Sutton,et al.  Mutation rates of the (1+1)-EA on pseudo-boolean functions of bounded epistasis , 2011, GECCO '11.

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

[22]  Carlo Mannino,et al.  Models and solution techniques for frequency assignment problems , 2003, 4OR.