## 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 (μ+1) GA solves arbitrary length-n instances of closest string in 2O(d2+dlogk)·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≤ t(n) ≤ poly (n) is the number of iterations allocated to the (μ+1) GA before a restart, which can be an arbitrary polynomial in n. This confirms that the multi-start (μ+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^{Ω}^{(}^{log}^{(}^{d}^{+}^{k}^{)}^{)} steps in expectation. The lower bound asserts that crossover is a necessary component in the FPT running time.

Original language | English (US) |
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Pages (from-to) | 1138-1163 |

Number of pages | 26 |

Journal | Algorithmica |

Volume | 83 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Copyright:

Copyright 2021 Elsevier B.V., All rights reserved.

## Keywords

- Combinatorial optimization
- Crossover
- Fixed-parameter tractability
- Runtime analysis