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

Leonid Gurvits*, Alex Samorodnitsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

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.

Original languageAmerican English
Pages (from-to)531-550
Number of pages20
JournalDiscrete and Computational Geometry
Volume27
Issue number4
DOIs
StatePublished - Jun 2002
Externally publishedYes

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