Approximating mixed volumes to arbitrary accuracy

We study the problem of approximating the mixed volume $V(P_1^{(\alpha_1)}, \dots, P_k^{(\alpha_k)})$ of an $k$-tuple of convex polytopes $(P_1, \dots, P_k)$, each of which is defined as the convex hull of at most $m_0$ points in $\mathbb{Z}^n$. We design an algorithm that produces an estimate that is within a multiplicative $1 \pm \epsilon$ factor of the true mixed volume with a probability greater than $1 - \delta.$ Let the constant $\prod_{i=2}^{k} \frac{(\alpha_{i}+1)^{\alpha_{i}+1}}{\alpha_{i}^{\,\alpha_{i}}}$ be denoted by $\tilde{A}$. When each $P_i \subseteq B_\infty(2^L)$, we show in this paper that the time complexity of the algorithm is bounded above by a polynomial in $n, m_0, L, \tilde{A}, \epsilon^{-1}$ and $\log \delta^{-1}$. In fact, a stronger result is proved in this paper, with slightly more involved terminology. In particular, we provide the first randomized polynomial time algorithm for computing mixed volumes of such polytopes when $k$ is an absolute constant, but $\alpha_1, \dots, \alpha_k$ are arbitrary. Our approach synthesizes tools from convex optimization, the theory of Lorentzian polynomials, and polytope subdivision.