Combinatorica

, Volume 4, Issue 4, pp 373–395 | Cite as

A new polynomial-time algorithm for linear programming

  • N. Karmarkar
Article

Abstract

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n3.5L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time.

AMS subject classification (1980)

90 C 05 

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References

  1. [1]
    H. S. M. Coxeter,Introduction to Geometry, Wiley (1961).Google Scholar
  2. [2]
    G. B. Dantzig,Linear Programming and Extensions, Princeton University Press, Princeton, NJ (1963).MATHGoogle Scholar
  3. [3]
    M. Grötschel, L. Lovász andA. Schrijver, The Ellipsoid Method and its Consequences in Combinatorial Optimization,Combinatorica 1 (1981), 169–197.MATHMathSciNetGoogle Scholar
  4. [4]
    L. G. Khachiyan, A polynomial Algorithm in Linear Programming,Doklady Akademii Nauk SSSR 244:S (1979), 1093–1096, translated inSoviet Mathematics Doklady 20:1 (1979), 191–194.MATHMathSciNetGoogle Scholar
  5. [5]
    V. Klee andG. L. Minty, How good is the simplex algorithm? inInequalities III, (ed. O. Shisha) Academic Press, New York, 1972, 159–179.Google Scholar
  6. [6]
    O. Veblen andJ. W. Young,Projective Geometry, 1–2, Blaisdell, New York, (1938).Google Scholar
  7. [7]
    R. J. Walker,Algebraic Curves, Princeton University Press (1950).Google Scholar

Copyright information

© Akadémiai Kiadó 1984

Authors and Affiliations

  • N. Karmarkar
    • 1
  1. 1.AT&T Bell LaboratoriesMurray HillU.S.A.

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