On the complexity of diophantine geometry in low dimensions (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
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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 of zonotopes is $\#\P$-hard (while each corresponding mixed volume can be computed easily) but also give examples showing that computing mixed volumes is hard even when computing the volume is easy. On the positive side, we derive a randomized algorithm for computing the mixed volumes $$ V(\overbrace{K_1\ld K_1}^{m_1}, \overbrace{K_2,\dots,K_2}^{m_2},\dots,\overbrace{K_s,\dots,K_s}^{m_s}) $$ of well-presented convex bodies $K_1,\dots,K_s$, where $m_1,\dots,m_s \in \N_0$ and $m_1 \geq n-\psi(n)$ with $\psi(n)=o(\frac{\log n}{\log \log n})$. The algorithm is an interpolation method based on polynomial-time randomized algorithms for computing the volume of convex bodies. This paper concludes with applications of our results to various problems in discrete mathematics, combinatorics, computational convexity, algebraic geometry, geometry of numbers, and operations research.