TY - JOUR

T1 - A deterministic algorithm for approximating the mixed discriminant and mixed volume, and a combinatorial corollary

AU - Gurvits, Leonid

AU - Samorodnitsky, Alex

PY - 2002/6

Y1 - 2002/6

N2 - We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n-tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in Rn to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A "side benefit" is a generalization of Rado's theorem on the existence of a linearly independent transversal.

AB - We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n-tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in Rn to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A "side benefit" is a generalization of Rado's theorem on the existence of a linearly independent transversal.

UR - http://www.scopus.com/inward/record.url?scp=0035999219&partnerID=8YFLogxK

U2 - 10.1007/s00454-001-0083-2

DO - 10.1007/s00454-001-0083-2

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AN - SCOPUS:0035999219

SN - 0179-5376

VL - 27

SP - 531

EP - 550

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -