### Abstract

We introduce and analyze a new technique Linear Programming Based Restriction (LP Restriction) for solving translational containment and enclosure problems. The containment task is to translate k m-gons into a n-gon container. The enclosure task is to translate k m-gons into a minimum area n-gon which is convex with fixed orientation edges. All running times are based on an assumption of fixed k and are asymptotic in m and n. Lower bounds are proved for containment in a nonconvex container: Ω((mn)^{2k}) for nonconvex polygons and Ω(n^{k}) for convex polygons. LP restriction achieves upper bound O((m^{2} + mn)^{2k} log n) for nonconvex polygons and O((mn)^{k} log n) for convex polygons. The former almost matches the lower bound, and the latter does also for constant m. For arbitrary m and k > 3, the latter is faster than any other known algorithm for translational containment of convex polygons. A proof is given that the area function for fixed-angle convex polygons is Lorentzian. Enclosure algorithms based on this result have running times O((m^{2} + n)^{2k-2}(n + log m)) for nonconvex polygons and O((m + n)^{2k}(n + log m)) or O(m^{k-1}(n^{2k+1} + log m)) for convex polygons. We are not aware of any previous running times for k-polygon minimum area enclosure. Other containment and enclosure results are given. LP restriction demonstrates a useful combination of mathematical programming and computational geometry which may be a paradigm for solving other tasks. We show that the containment and enclosure algorithms are useful in industry. Software based on LP restriction has been licensed to industry, and it is the fastest available solution to translational containment and enclosure for nonconvex polygons.

Original language | English (US) |
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Title of host publication | Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |

Publisher | Association for Computing Machinery |

Pages | 109-118 |

Number of pages | 10 |

ISBN (Electronic) | 0897917855 |

DOIs | |

State | Published - Jul 1 1996 |

Event | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 - Philadelphia, United States Duration: May 22 1996 → May 24 1996 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F129452 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |
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Country | United States |

City | Philadelphia |

Period | 5/22/96 → 5/24/96 |

### ASJC Scopus subject areas

- Software

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## Cite this

*Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996*(pp. 109-118). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F129452). Association for Computing Machinery. https://doi.org/10.1145/237814.237840