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 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.