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A Subexponential Bound for Linear Programming (1996)
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| Venue: | ALGORITHMICA |
| Citations: | 184 - 15 self |
BibTeX
@MISC{Matousek96asubexponential,
author = {Jiri Matousek and Micha Sharir and Emo Welzl},
title = { A Subexponential Bound for Linear Programming },
year = {1996}
}
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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).
Keyphrases
linear programming subexponential bound linear program internal randomization linear inequality subexponential running time smallest nonnegative point linear programming algorithm simple randomized algorithm expected bound abstract framework arithmetic operation unit cost model recent result
