Algorithmica

, Volume 16, Issue 4–5, pp 498–516

A subexponential bound for linear programming

  • J. Matoušek
  • M. Sharir
  • E. Welzl
Article

Abstract

We present a simple randomized algorithm which solves linear programs withn constraints andd variables in expected
$$\min \{ O(d^2 2^d n),e^{2\sqrt {dIn({n \mathord{\left/ {\vphantom {n {\sqrt d }}} \right. \kern-\nulldelimiterspace} {\sqrt d }})} + O(\sqrt d + Inn)} \}$$
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 satisfyingn given linear inequalities ind 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
$$O(d^2 n + e^{O(\sqrt {dInd} )} ).$$

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) ofn points ind-space, computing the distance of twon-vertex (orn-facet) polytopes ind-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).

Key words

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

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Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • J. Matoušek
    • 1
  • M. Sharir
    • 2
    • 3
  • E. Welzl
    • 4
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czechoslovakia
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.Institut für theoretische InformatikETH ZürichZürichSwitzerland

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