How to compute the volume in low dimension?

2025-03-04

AI Summary

This work addresses the high-precision query complexity of estimating the volume of convex bodies in constant dimension $d$. For deterministic, randomized, and quantum query models, it establishes the first tight complexity characterization in low dimensions—breaking from classical asymptotic analyses focused on high-dimensional regimes. The approach integrates tools from convex geometry, probabilistic methods, random walk theory, and quantum query modeling: it constructs optimal decision trees and develops a novel quantum lower bound technique. The results show that, as the additive error $varepsilon o 0$, the query complexity for all three models is $Theta(log(1/varepsilon))$, with matching upper and lower bounds up to constant factors. This resolves a long-standing open problem on query complexity bounds for convex body volume estimation and yields the first exact complexity theory for volume estimation in constant dimensions.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

Problem

• Estimating volume of convex bodies in low dimensions.
• Determining query complexity for volume estimation.
• Characterizing deterministic, randomized, and quantum query complexities.

Innovation

• Deterministic query complexity in low dimension
• Randomized algorithms for volume estimation
• Quantum query complexity in constant dimension