Approximating mixed volumes to arbitrary accuracy

TL;DR

This paper introduces a randomized polynomial-time algorithm for approximating the mixed volume of convex polytopes with arbitrary parameters, combining convex optimization, Lorentzian polynomials, and polytope subdivision techniques.

Contribution

It provides the first efficient randomized algorithm for mixed volume approximation with arbitrary parameters when the number of polytopes is fixed.

Findings

Algorithm achieves a multiplicative approximation within 1 ± ε with high probability.

Time complexity is polynomial in key parameters including dimension, size, and approximation factors.

First known polynomial-time method for mixed volume approximation under these conditions.

Abstract

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.