A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING (1984)
| Venue: | COMBINATORICA |
| Citations: | 850 - 3 self |
BibTeX
@MISC{Karmarkar84anew,
author = {N. Karmarkar},
title = { A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING},
year = {1984}
}
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Abstract
We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requires O(tf'SL) arithmetic operations on O(L) bit numbers, where n is the number of variables and L is the number of bits in the input. The running,time of this algorithm is better than the ellipsoid algorithm by a factor of O(n~'~). We prove that given a polytope P and a strictly in-terior point a E P, there is a projective transformation of the space that maps P, a to P', a ' having the following property. The ratio of the radius of the smallest sphere with center a', containing P ' to the radius of the largest sphere with center a ' contained in P ' is O(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 poly-nomial time.
