How often should you clean your room?

TL;DR

This paper investigates a combinatorial optimization problem related to cleaning frequency, providing asymptotic analysis, bounds, and conjectures for the optimal cleanup time based on the number of objects.

Contribution

It introduces a new combinatorial problem motivated by cleaning frequency and offers asymptotic bounds, approximate solutions, and conjectures for the optimal cleanup time.

Findings

Proved a logarithmic upper bound for cleanup time

Solved an approximate version of the problem

Conjectured a precise logarithmic asymptotic

Abstract

We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic.