Approximating mixed volumes to arbitrary accuracy
Hariharan Narayanan, Sourav Roy
TL;DR
This work presents a randomized polynomial-time algorithm to approximate the mixed volume $V(P_1^{(\alpha_1)}, \dots, P_k^{(\alpha_k)})$ for polytopes $P_i$ defined as the convex hull of at most $m_0$ lattice points. The approach combines a generalized Gurvits-capacity surrogate (via $\operatorname{Cap}_{\alpha}(p)$ and Lorentzian polynomial theory), Schneider’s subdivision of the Minkowski sum, and a sampling-based estimator with Chernoff bounds to achieve a $(1\pm\epsilon)$-approximation with probability $>1-\delta$; the runtime is polynomial in $n,m_0,L,A,\epsilon^{-1}$ and $\log\delta^{-1}$ when $P_i\subseteq B_\infty(2^L)$ and $k$ is fixed. The key contributions include establishing capacity-to-mixed-volume bounds through Lorentzian polynomials, a practical subdivision theorem to decompose the mixed-volume into face contributions, and a concrete randomized algorithm (Algorithm 1) whose complexity improves upon prior results via a refined constant $A$ (with $A \le \tilde{A}$). The results advance efficient computation of mixed volumes in this polytope regime, bridging convex optimization, Lorentzian polynomial theory, and polytope subdivision to enable scalable applications in algebraic geometry and related fields.
Abstract
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.