Approximating mixed volumes to arbitrary accuracy

2025-08-27

AI Summary

This paper addresses the efficient approximation of the mixed volume (V(P_1^{alpha_1},dots,P_k^{alpha_k})) of (k) integer convex polytopes in (mathbb{R}^n), each defined as the convex hull of at most (m_0) lattice points. For constant (k), we present the first randomized polynomial-time algorithm achieving a ((1pmvarepsilon))-approximation with probability at least (1-delta), for arbitrary exponents (alpha_i). Our method integrates Lorentzian polynomial theory, convex optimization, and polyhedral subdivision techniques to construct a unified approximation framework. The algorithm runs in time (mathrm{poly}(n, m_0, L, ilde{A}, varepsilon^{-1}, log delta^{-1})), where (L) is the bit-length of input coordinates and ( ilde{A}) bounds the polytopes’ diameters. This work overcomes prior limitations—where mixed volume approximation was restricted to small fixed (k) or special polytope classes—and delivers the first general-purpose, efficient, and provably accurate randomized approximation scheme for this fundamental problem in algebraic and convex geometry.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $ ilde{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 $n, m_0, L, ilde{A}, ε^{-1}$ 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.

Problem

• Approximating mixed volumes of convex polytopes with arbitrary accuracy
• Developing randomized polynomial-time algorithm for mixed volume computation
• Handling arbitrary exponents while maintaining computational efficiency

Innovation

• Randomized polynomial time mixed volume algorithm
• Multiplicative approximation with confidence bounds
• Synthesizes convex optimization and Lorentzian polynomials