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Fixed-Parameter Tractability of Crossover: Steady-State GAs on the Closest String Problem

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Abstract

We investigate the effect of crossover in the context of parameterized complexity on a well-known fixed-parameter tractable combinatorial optimization problem known as the closest string problem. We prove that a multi-start (\(\mu\)+1) GA solves arbitrary length-n instances of closest string in \(2^{O(d^2 + d \log k)} \cdot t(n)\) steps in expectation. Here, k is the number of strings in the input set, d is the value of the optimal solution, and \(n \le t(n) \le {\text {poly}}(n)\) is the number of iterations allocated to the (\(\mu\)+1) GA before a restart, which can be an arbitrary polynomial in n. This confirms that the multi-start (\(\mu\)+1) GA runs in randomized fixed-parameter tractable (FPT) time with respect to the above parameterization. On the other hand, if the crossover operation is disabled, we show there exist instances that require \(n^{\varOmega (\log (d+k))}\) steps in expectation. The lower bound asserts that crossover is a necessary component in the FPT running time.

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Notes

  1. Note that the claimed bound is trivial when \(|{\text {supp}}({\varvec{x}})| \ge \ell\).

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    Andrew M. Sutton

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Sutton, A.M. Fixed-Parameter Tractability of Crossover: Steady-State GAs on the Closest String Problem. Algorithmica 83, 1138–1163 (2021). https://doi.org/10.1007/s00453-021-00809-8

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