# A new polynomial-time algorithm for linear programming

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DOI: 10.1007/BF02579150

- Cite this article as:
- Karmarkar, N. Combinatorica (1984) 4: 373. doi:10.1007/BF02579150

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## Abstract

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requires*O*(*n*^{3.5}*L*) 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*^{2.5}). We prove that given a polytope*P* and a strictly interior point a ε*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 polynomial time.