### 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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### 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

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*Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11088 LNCS, Springer Verlag, pp. 392-403, 22nd International Conference on Developments in Language Theory, DLT 2018, Tokyo, Japan, 18/9/10. https://doi.org/10.1007/978-3-319-98654-8_32

**Closest Substring Problems for Regular Languages.** / Han, Yo Sub; Ko, Sang Ki; Ng, Timothy; Salomaa, Kai.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Closest Substring Problems for Regular Languages

AU - Han, Yo Sub

AU - Ko, Sang Ki

AU - Ng, Timothy

AU - Salomaa, Kai

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85053888367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053888367&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-98654-8_32

DO - 10.1007/978-3-319-98654-8_32

M3 - Conference contribution

AN - SCOPUS:85053888367

SN - 9783319986531

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 392

EP - 403

BT - Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings

A2 - Hoshi, Mizuho

A2 - Seki, Shinnosuke

PB - Springer Verlag

ER -