TY - GEN

T1 - A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume

AU - Gurvits, Leonid

AU - Samorodnitsky, Alex

PY - 2000

Y1 - 2000

N2 - We present a deterministic polynomial algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within a multiplicative factor of en. 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. We obtain tight upper and lower bounds on the mixed discriminant of doubly stochasic n-tuples, proving a conjecture of Bapat, and generalizing the van der Waerden - Falikman - Egorychev theorem. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in Rn to within a multiplicative factor of nO(n). This answers a question of Dyer, Gritzmann and Hufnagel.

AB - We present a deterministic polynomial algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within a multiplicative factor of en. 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. We obtain tight upper and lower bounds on the mixed discriminant of doubly stochasic n-tuples, proving a conjecture of Bapat, and generalizing the van der Waerden - Falikman - Egorychev theorem. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in Rn to within a multiplicative factor of nO(n). This answers a question of Dyer, Gritzmann and Hufnagel.

UR - http://www.scopus.com/inward/record.url?scp=0033727271&partnerID=8YFLogxK

U2 - 10.1145/335305.335311

DO - 10.1145/335305.335311

M3 - Conference contribution

AN - SCOPUS:0033727271

SN - 1581131844

SN - 9781581131840

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 48

EP - 57

BT - Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC 2000

T2 - 32nd Annual ACM Symposium on Theory of Computing, STOC 2000

Y2 - 21 May 2000 through 23 May 2000

ER -