We study the problem of approximating the mixed volume $V(P_1^{(α_1)}, \dots, P_k^{(α_k)})$ 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 $1 \pm ε$ factor of the true mixed volume with a probability greater than $1 - δ.$ Let the constant $ \prod_{i=2}^{k} \frac{(α_{i}+1)^{α_{i}+1}}{α_{i}^{\,α_{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 $n, m_0, L, \tilde{A}, ε^{-1}$ and $\log δ^{-1}$. 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 $α_1, \dots, α_k$ are arbitrary. Our approach synthesizes tools from convex optimization, the theory of Lorentzian polynomials, and polytope subdivision.

翻译：我们研究逼近由至多 $m_0$ 个 $\mathbb{Z}^n$ 中的点定义的凸多面体 $(P_1, \dots, P_k)$ 的混合体积 $V(P_1^{(α_1)}, \dots, P_k^{(α_k)})$ 的问题。我们设计了一种算法，能以大于 $1 - δ$ 的概率生成与真实混合体积的误差在 $1 \pm ε$ 倍乘因子内的估计值。令常数 $ \prod_{i=2}^{k} \frac{(α_{i}+1)^{α_{i}+1}}{α_{i}^{\,α_{i}}}$ 记为 $\tilde{A}$。当每个 $P_i \subseteq B_\infty(2^L)$ 时，本文证明该算法的时间复杂度上界为 $n, m_0, L, \tilde{A}, ε^{-1}$ 和 $\log δ^{-1}$ 的多项式。实际上，本文证明了一个更强结果，其术语表述稍显复杂。特别地，当 $k$ 为绝对常数而 $α_1, \dots, α_k$ 任意时，我们首次提出了计算此类多面体混合体积的随机多项式时间算法。我们的方法综合了凸优化、洛伦兹多项式理论和多面体细分等工具。