A tight lower bound for the online bounded space hypercube bin packing problem

TL;DR

This paper establishes that the asymptotic performance ratio for online bounded space hypercube bin packing in d dimensions is tightly bounded by d/\u221a{ ext{log}d}, confirming a conjecture and refining previous bounds.

Contribution

The paper proves that the asymptotic performance ratio is d/{ ext{log}d}, providing a tight bound and resolving a conjecture.

Findings

The performance ratio is d/{ ext{log}d}

The ratio is tightly bounded, confirming the conjecture

Uses probabilistic methods for the proof

Abstract

In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $\rho$ of the online bounded space variant is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$, using probabilistic arguments.