, Volume 4, Issue 4, pp 373395
First online:
A new polynomialtime algorithm for linear programming
 N. KarmarkarAffiliated withAT&T Bell Laboratories
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We present a new polynomialtime algorithm for linear programming. In the worst case, the algorithm requiresO(n ^{3.5} L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The runningtime of this algorithm is better than the ellipsoid algorithm by a factor ofO(n ^{2.5}). 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 Title
 A new polynomialtime algorithm for linear programming
 Journal

Combinatorica
Volume 4, Issue 4 , pp 373395
 Cover Date
 198412
 DOI
 10.1007/BF02579150
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 90 C 05
 Industry Sectors
 Authors

 N. Karmarkar ^{(1)}
 Author Affiliations

 1. AT&T Bell Laboratories, 07974, Murray Hill, NJ, U.S.A.