Year: 2026
Type: preprint
Abstract: We study the computation of mixed volumes of convex sets given as rational subspace sections of bodies described by membership oracles. The sets need not have nonempty interior. For positive multiplicities $α_1+\cdots+α_k=n$, we give a randomized algorithm that computes a relative $\varepsilon$-approximation with probability at least $1-δ$. Its running time is polynomial in Gurvits' factor $Γ_α=(n^n/n!)\prod_iα_i!/α_i^{α_i}$, the input presentation length, and $\varepsilon^{-1}$, and linear in $1+\logδ^{-1}$, with degree independent of $k$. The running-time bound holds for every choice of the random bits, and zero mixed volumes are detected exactly. We extend the capacity-scaling and sampling method of Narayanan--Roy by using rational projections and a smooth mixed Monge--Ampère representation. In particular, the algorithm runs in randomized polynomial time when the number of distinct convex sets is bounded above by a constant.
Source: arXiv (Cornell University)
Author: Narayanan, Hariharan
Institution: Tata Institute of Fundamental Research
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G240064222
OpenAlex ID: W7221002721