### Abstract

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ1 and γ2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ1 and γ2 where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems.

We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fréchet Distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.

Original language | English |
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Title of host publication | Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 7-10 January 2018, New Orleans, Louisiana |

Place of Publication | New York |

Publisher | Association for Computing Machinery, Inc |

Pages | 1121-1134 |

Number of pages | 14 |

ISBN (Electronic) | 978-1-61197-503-1 |

DOIs | |

Publication status | Published - 7 Jan 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018) - Astor Crowne Plaza, New Orleans, United States Duration: 7 Jan 2018 → 10 Jan 2018 Conference number: 29 https://www.siam.org/meetings/da18/ |

### Conference

Conference | 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018) |
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Abbreviated title | SODA 2018 |

Country | United States |

City | New Orleans |

Period | 7/01/18 → 10/01/18 |

Internet address |

### Fingerprint

### Cite this

*Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 7-10 January 2018, New Orleans, Louisiana*(pp. 1121-1134). New York: Association for Computing Machinery, Inc. https://doi.org/10.1137/1.9781611975031.73

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*Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 7-10 January 2018, New Orleans, Louisiana.*Association for Computing Machinery, Inc, New York, pp. 1121-1134, 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), New Orleans, United States, 7/01/18. https://doi.org/10.1137/1.9781611975031.73

**On the complexity of optimal homotopies.** / Chambers, E.W.; de Mesmay, A.; Ophelders, T.A.E.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

TY - GEN

T1 - On the complexity of optimal homotopies

AU - Chambers, E.W.

AU - de Mesmay, A.

AU - Ophelders, T.A.E.

PY - 2018/1/7

Y1 - 2018/1/7

N2 - In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ1 and γ2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ1 and γ2 where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems.We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fréchet Distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.

AB - In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ1 and γ2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ1 and γ2 where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems.We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fréchet Distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.

U2 - 10.1137/1.9781611975031.73

DO - 10.1137/1.9781611975031.73

M3 - Conference contribution

SP - 1121

EP - 1134

BT - Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 7-10 January 2018, New Orleans, Louisiana

PB - Association for Computing Machinery, Inc

CY - New York

ER -