## Abstract

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Fréchet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Fréchet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fréchet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an eO((n+m)^{6} min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n + m)^{4}) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.

Original language | English |
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Title of host publication | 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016 |

Editors | Anca Muscholl, Piotr Faliszewski, Rolf Niedermeier |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770163 |

DOIs | |

State | Published - 1 Aug 2016 |

Externally published | Yes |

Event | 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016 - Krakow, Poland Duration: 22 Aug 2016 → 26 Aug 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 58 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016 |
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Country/Territory | Poland |

City | Krakow |

Period | 22/08/16 → 26/08/16 |

### Bibliographical note

Publisher Copyright:© Yuta Fujishige, Yuki Tsujimaru, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda.

## Keywords

- Chain simplification
- Discrete Fréchet distance
- Dynamic programming
- Geometric arrangements
- Protein structural resemblance