Optimal sequencing of a set of positive numbers with the variance of the sequence's partial sums maximized

TL;DR

This paper investigates the optimal arrangement of positive numbers to maximize the variance of partial sums, revealing a structured solution and contrasting it with the NP-complete problem of minimizing variance.

Contribution

It introduces a novel approach to sequence optimization for variance maximization and characterizes the structure of the optimal sequence.

Findings

Optimal sequence has a specific, elegant structure.

Maximizing variance differs fundamentally from minimizing variance.

The problem's structure offers insights into sequence optimization.

Abstract

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.