Online Two-Dimensional Vector Packing with Advice

TL;DR

This paper studies the online two-dimensional vector packing problem, establishing bounds on competitive ratios for strategies with advice, and introduces new strategies with improved ratios for specific vector angle ranges.

Contribution

It provides new strategies with competitive ratios for vector packing with advice, including bounds for restricted angles and an improved ratio for unrestricted vectors.

Findings

Lower bound of 11/5 on competitive ratio for any { extsc{AnyFit}} strategy.

Strategies with competitive ratio depending on vector angles and advice.

A 5/2-competitive strategy for unrestricted vectors.

Abstract

We consider the online two-dimensional vector packing problem, showing a lower bound of $11/5$ on the competitive ratio of any {\sc AnyFit} strategy for the problem. We provide strategies with competitive ratio $\max\!\left\{2,6\big/\big(1+3\tan(\pi/4-\gamma/2)\big)+\epsilon\right\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi/4-\gamma/2,\pi/4+\gamma/2]$, for $0\leq\gamma<\pi/3$ and $\max\left\{5/2,4\big/\big(1+2\tan(\pi/4-\gamma/2)\big)+\epsilon\right\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi/4-\gamma/2,\pi/4+\gamma/2]$, for $0\leq\gamma\leq\pi/3$. In addition, we give a $5/2$-competitive strategy also using logarithmic advice for the unrestricted vectors case. These results should be contrasted to the currently best competitive strategy, FirstFit, having competitive ratio~$27/10$.