The analysis of evolutionary algorithms on sorting and shortest paths problems

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Abstract

The analysis of evolutionary algorithms is up to now limited to special classes of functions and fitness landscapes. E.g., it is not possible to characterize the set of TSP instances (or another NP-hard combinatorial optimization problem) which are solved by a generic evolutionary algorithm (EA) in an expected time bounded by some given polynomial. As a first step from artificial functions to typical problems from combinatorial optimization, we analyze simple EAs on well-known problems, namely sorting and shortest paths. Although it cannot be expected that EAs outperform the well-known problem specific algorithms on these simple problems, it is interesting to analyze how EAs work on these problems. The following results are obtained:

  • - Sorting is the maximization of “sortedness” which is measured by one of several well-known measures of presortedness. The different measures of presortedness lead to fitness functions of quite different difficulty for EAs.

  • - Shortest paths problems are hard for all types of EA, if they are considered as single-objective optimization problems, whereas they are easy as multi-objective optimization problems.

  • This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).