We present a new polynomial-time algorithm for linear programming• The running-time of this algorithm is O(n3"SL2), as compared to O(n6L 2) for the ellipsoid algorithm. We prove that given a polytope P and a strictly interior 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 polynomial-time.
Some Comments on the Significance of the Result
Worst-case Bounds on Linear ProgrammingThe simplex algorithm for linear programming has been shown to require an exponential number of steps in the worst-case [1]. A polynomial-time algorithm for linear programming was published by Khachiyan in 1979 [2].The complexity of this algorithm is O(n6L 2) where n is the ~mension of the problem and L is the number of bits in the input [3]. In this paper we present a new polynomial-time algorithm for linear programming whose timecomplexity is O (n3"SL2).
Polytopes and Projective GeometryWe prove a theorem about polytopes which seems to be interesting in its own right. Given a polytope P ~ R n and a strictly interior 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' in O(n).
In this paper, we consider computations involving polynomials with inexact coefficients, i.e. with bounded coefficient errors. The presence of input errors changes the nature of questions traditionally asked in computer algebra. For instance, given two polynomials, instead of trying to compute their greatest common divisor, one might now try to compute a pair of polynomials with a non-trivial common divisor close to the input polynomials. We consider the problem of finding approximate common divisors in the context of inexactly specified polynomials. We develop efficient algorithms for the so-called nearest common divisor problem and several of its variants.
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