Improved Runtime Bounds for the (1+1) EA on Random 3-CNF Formulas Based on Fitness-Distance Correlation

With this paper, we contribute to the theoretical understanding of randomized search heuristics by investigating their behavior on random 3-SAT instances. We improve the results for the (1+1) EA obtained by Sutton and Neumann [PPSN 2014, 942--951] in three ways. First, we reduce the upper bound by a linear factor and prove that the (1+1) EA obtains optimal solutions in time $O(n \log n)$ with high probability on asymptotically almost all high-density satisfiable 3-CNF formulas. Second, we extend the range of densities for which this bound holds to satisfiable formulas of at least logarithmic density. Finally, we complement these mathematical results with numerical experiments that summarize the behavior of the (1+1) EA on formulas along the density spectrum, and suggest that the implicit constants hidden in our bounds are low. Our proofs are based on analyzing the run of the algorithm by establishing a fitness-distance correlation. This approach might be of independent interest and we are optimistic that it is useful for the analysis of randomized search heuristics in various other settings. To our knowledge, this is the first time that fitness-distance correlation is explicitly used to rigorously prove a performance statement for an evolutionary algorithm.

[1]  Michael Molloy,et al.  Cores in random hypergraphs and Boolean formulas , 2005, Random Struct. Algorithms.

[2]  Benjamin Doerr,et al.  Multiplicative Drift Analysis , 2010, GECCO '10.

[3]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods on Classifications of Fitness Functions on Classifications of Fitness Functions , 2022 .

[4]  Michael Krivelevich,et al.  Solving random satisfiable 3CNF formulas in expected polynomial time , 2006, SODA '06.

[5]  Hector J. Levesque,et al.  Hard and Easy Distributions of SAT Problems , 1992, AAAI.

[6]  Dimitris Achlioptas,et al.  Random Satisfiability , 2009, Handbook of Satisfiability.

[7]  Carsten Witt,et al.  Bioinspired Computation in Combinatorial Optimization , 2010, Bioinspired Computation in Combinatorial Optimization.

[8]  Tobias Storch,et al.  Finding large cliques in sparse semi-random graphs by simple randomized search heuristics , 2007, Theor. Comput. Sci..

[9]  M. Mitzenmacher.  Tight Thresholds for The Pure Literal Rule , 1997 .

[10]  Frank Neumann,et al.  Bioinspired computation in combinatorial optimization: algorithms and their computational complexity , 2010, GECCO '12.

[11]  Thomas Jansen,et al.  Analyzing Evolutionary Algorithms: The Computer Science Perspective , 2012 .

[12]  Jeanette P. Schmidt,et al.  Component structure in the evolution of random hypergraphs , 1985, Comb..

[13]  Gabriel Istrate,et al.  CRITICAL BEHAVIOR IN THE SATISFIABILITY OF RANDOM K-HORN FORMULAE , 2007 .

[14]  James M. Crawford,et al.  Experimental Results on the Crossover Point inSatis ability , 1993 .

[15]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

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

[17]  Eli Ben-Sasson,et al.  Linear Upper Bounds for Random Walk on Small Density Random 3-CNFs , 2007, SIAM J. Comput..

[18]  Lee Altenberg,et al.  Fitness Distance Correlation Analysis: An Instructive Counterexample , 1997, ICGA.

[19]  Leslie Ann Goldberg,et al.  Adaptive Drift Analysis , 2011, Algorithmica.

[20]  Dan Gutfreund,et al.  Finding a randomly planted assigment in a random 3CNF , 2002 .

[21]  Anne Auger,et al.  Theory of Randomized Search Heuristics: Foundations and Recent Developments , 2011, Theory of Randomized Search Heuristics.

[22]  Carsten Witt,et al.  Fitness levels with tail bounds for the analysis of randomized search heuristics , 2014, Inf. Process. Lett..

[23]  Victor J. Rayward-Smith,et al.  Fitness Distance Correlation and Ridge Functions , 1998, PPSN.

[24]  Frank Neumann,et al.  Runtime Analysis of Evolutionary Algorithms on Randomly Constructed High-Density Satisfiable 3-CNF Formulas , 2014, PPSN.

[25]  Alan M. Frieze,et al.  Analysis of Two Simple Heuristics on a Random Instance of k-SAT , 1996, J. Algorithms.

[26]  Amin Coja-Oghlan,et al.  On the solution‐space geometry of random constraint satisfaction problems , 2011, Random Struct. Algorithms.

[27]  Thomas Jansen,et al.  Analyzing Evolutionary Algorithms , 2015, Natural Computing Series.