## A Subexponential Bound for Linear Programming (1996)

Venue: | ALGORITHMICA |

Citations: | 164 - 16 self |

### BibTeX

@MISC{Matousek96asubexponential,

author = {Jiri Matousek and Micha Sharir and Emo Welzl},

title = { A Subexponential Bound for Linear Programming },

year = {1996}

}

### Years of Citing Articles

### OpenURL

### 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).