## Abstract

We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d^{2}2^{d}n), e^{2√d ln(n√d)+O(√d+ln n}} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of O(d^{2}n + e^{O(√d ln d)}). The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in d-space, computing the distance of two n-vertex (or n-facet) polytopes in d-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).

Original language | English |
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Pages (from-to) | 498-516 |

Number of pages | 19 |

Journal | Algorithmica |

Volume | 16 |

Issue number | 4-5 |

DOIs | |

State | Published - 1996 |

## Keywords

- Combinatorial optimization
- Computational geometry
- Linear programming
- Randomized incremental algorithms
- Smallest enclosing ball
- Smallest enclosing ellipsoid