Watson, Richard A. “Analysis of recombinative algorithms on a non-separable building-block problem.” In *Foundations of Genetic Algorithms 6* , pp. 69-89. Morgan Kaufmann, 2001.

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Our analysis seeks to exemplify the utility of crossover by studying a nonseparable building-block problem that is as easy as possible under recombination but very hard for any kind of mutation-based algorithm. The interdependent fitness contributions of blocks in this problem produce a search space that has an exponential number of local optima for a mutation-only algorithm. In contrast, the problem has no local optima for a recombinative algorithm - that is, there is always a path of monotonically increasing fitness leading to the global optimum. We give an upper bound on the expected time for a recombinative algorithm to solve this problem by proving the existence of a path to the solution and calculating the time for each step on this path. Ordinarily, such a straightforward approach would be defeated because both the existence of a path, and the time for a step, are dependent on the state of the population when using recombination. However, to calculate an upper bound on the expected time it is sufficient to know certain properties, or invariants, of the population rather than its exact state. Our initial proofs utilise a ‘recombinative hill-climber’, which applies crossover repeatedly to just two strings, for this purpose. Though our analysis does not transfer directly to a GA with a full population, a solution time based on the assumption that the population has the necessary invariant properties agrees with empirical results.