### Abstract

It is well known that given a finite set of strings of equal length, the Consensus String problem—the problem of deciding whether or not there exists a consensus string whose distance is at most r from every string in the given set—is proven to be NP-complete. A similar problem called the Closest Substring problem asks whether there exists a string w of length l such that each string in a given set L has a substring whose distance is at most r (called radius) from w. As the Closest Substring problem is a generalized version of the Consensus String problem, it is obvious that the problem is NP-hard for a finite set of strings. We show that the Closest Substring problem for regular languages represented by nondeterministic finite automata (NFAs) is PSPACE-complete. The main difference from the previous work is that we consider an infinite set of strings, which is recognized by an NFA as input instead of a finite set of strings. We also prove that the Closest Substring problem for acyclic NFAs lies in the second level of the polynomial-time hierarchy (formula presented) and is both NP-hard and coNP-hard.

Original language | English |
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Title of host publication | Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings |

Editors | Mizuho Hoshi, Shinnosuke Seki |

Publisher | Springer Verlag |

Pages | 392-403 |

Number of pages | 12 |

ISBN (Print) | 9783319986531 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

Event | 22nd International Conference on Developments in Language Theory, DLT 2018 - Tokyo, Japan Duration: 2018 Sep 10 → 2018 Sep 14 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11088 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 22nd International Conference on Developments in Language Theory, DLT 2018 |
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Country | Japan |

City | Tokyo |

Period | 18/9/10 → 18/9/14 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings*(pp. 392-403). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11088 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-98654-8_32