## Abstract

We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick"or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n ^{7}) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1-ε)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ε)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).

Original language | English |
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Pages | 474-483 |

Number of pages | 10 |

DOIs | |

State | Published - 28 Feb 2006 |

Event | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms - Miami, FL, United States Duration: 22 Jan 2006 → 24 Jan 2006 |

### Conference

Conference | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | Miami, FL |

Period | 22/01/06 → 24/01/06 |

## ASJC Scopus subject areas

- Software
- General Mathematics