Computing Mixed Discriminants , Mixed Volumes

A characterization of the mixed discriminant

The striking analogy between mixed volumes of convex bodies and mixed discriminants of positive semidefinite matrices has repeatedly been observed. It was A. D. Aleksandrov [1] who, in his second proof of the Aleksandrov–Fenchel inequalities for the mixed volumes, first introduced the mixed discriminants of positive semidefinite quadratic forms and established some of their properties, includin...

An efficient tree decomposition method for permanents and mixed discriminants

We present an efficient algorithm to compute permanents, mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays (tensors). We describe the sparsity structure of an array in terms of a graph, and we assume that its treewidth, denoted as ω, is small. Our algorithm requires Õ(n 2) arithmetic operations to compute permanents, and Õ(n + n 3) for mixed discrimina...

On the Complexity of Computing Mixed Volumes

This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #P-hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes. We show that computing the volume o...

The Brunn-Minkowski-Type Inequality

Mixed Volume Computation in Parallel

Efficient algorithms for computing mixed volumes, via the computation of mixed cells, have been implemented in DEMiCs [18] and MixedVol-2.0 [13]. While the approaches in those two packages are somewhat different, they follow the same theme and are both highly serial. To fit the need for the parallel computing, a reformulation of the algorithms is inevitable. This article proposes a reformulatio...