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A subexponential bound for linear programming

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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).

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 11800, Praha 1, Czechoslovakia

    J. Matoušek

  2. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    M. Sharir

  3. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    M. Sharir

  4. Institut für theoretische Informatik, ETH Zürich, 8092, Zürich, Switzerland

    E. Welzl

Authors
  1. J. Matoušek
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Additional information

Communicated by M. Luby.

Work by the first author has been supported by a Humboldt Research Fellowship. Work by the second and third authors has been supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.). Work by the second author has been supported by Office of Naval Research Grant N00014-90-J-1284, by National Science Foundation Grants CCR-89-01484 and CCR-90-22103, and by grants from the U.S.-Israeli Binational Science Foundation, and the Fund for Basic Research administered by the Israeli Academy of Sciences. A preliminary version appeared inProc. 8th Annual ACM Symposium on Computational Geometry, 1992, pp. 1–8.

Work has been carried out while author was visiting the Institute for Computer Science, Berlin Free University.

Research has been carried out when the author was still at Freie Universität Berlin, Fachbereich Mathematik und Informatik.

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Matoušek, J., Sharir, M. & Welzl, E. A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996). https://doi.org/10.1007/BF01940877

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