### 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 | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Publisher | ACM |

Pages | 109-118 |

Number of pages | 10 |

State | Published - 1996 |

Event | Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing - Philadelphia, PA, USA Duration: May 22 1996 → May 24 1996 |

### Other

Other | Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing |
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City | Philadelphia, PA, USA |

Period | 5/22/96 → 5/24/96 |

### Fingerprint

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 109-118). ACM.

**Translational polygon containment and minimal enclosure using linear programming based restriction.** / Milenkovic, Victor.

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

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*ACM, pp. 109-118, Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing, Philadelphia, PA, USA, 5/22/96.

}

TY - GEN

T1 - Translational polygon containment and minimal enclosure using linear programming based restriction

AU - Milenkovic, Victor

PY - 1996

Y1 - 1996

N2 - 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(nk) for convex polygons. LP restriction achieves upper bound O((m2 + 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((m2 + n)2k-2 (n + log m)) for nonconvex polygons and O((m + n)2k (n + log m)) or O(mk-1 (n2k+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.

AB - 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(nk) for convex polygons. LP restriction achieves upper bound O((m2 + 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((m2 + n)2k-2 (n + log m)) for nonconvex polygons and O((m + n)2k (n + log m)) or O(mk-1 (n2k+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.

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M3 - Conference contribution

AN - SCOPUS:0029703214

SP - 109

EP - 118

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

PB - ACM

ER -