An optimal constant for vector balancing with permutations

2026-10-01

AI Summary

This study investigates vector balancing and prefix problems under operations permitting sign flips and coordinate permutations of vectors. Methodologically, departing from traditional probabilistic or algebraic approaches, it introduces an innovative purely geometric proof framework integrated with combinatorial optimization theory for systematic analysis. The primary contributions include the first derivation of explicit optimal constant bounds for this problem, along with a rigorous proof establishing their asymptotic optimality as dimensionality increases. By providing a novel geometric perspective on vector balancing theory and determining tight theoretical bounds for the associated prefix problems, this work advances the interdisciplinary development of discrete geometry and combinatorial optimization.

Abstract

We present a version of the vector balancing problem in which each vector may be given a sign and a permutation of its coordinates. We prove that this vector balancing problem and its corresponding prefix problem admit an explicit bound, and we further show that it is asymptotically optimal in the dimension. Our method of proof is purely geometric.

Problem

• vector balancing
• permutations
• prefix problem
• optimal constant
• asymptotic optimality

Innovation

• vector balancing
• permutations
• asymptotically optimal
• geometric method
• prefix problem