Online Two-Dimensional Vector Packing with Advice

We consider the online two-dimensional vector packing problem, showing a lower bound of $11\mathrm{/}5$ on the competitive ratio of any {\sc AnyFit} strategy for the problem. We provide strategies with competitive ratio $\max \text{\hspace{0.17em}⁣}\{2,6/(1+3\tan (\pi \mathrm{/}4-\gamma \mathrm{/}2))+ϵ\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi \mathrm{/}4-\gamma \mathrm{/}2,\pi \mathrm{/}4+\gamma \mathrm{/}2]$, for $0\le \gamma <\pi \mathrm{/}3$ and $\max \{5\mathrm{/}2,4/(1+2\tan (\pi \mathrm{/}4-\gamma \mathrm{/}2))+ϵ\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi \mathrm{/}4-\gamma \mathrm{/}2,\pi \mathrm{/}4+\gamma \mathrm{/}2]$, for $0\le \gamma \le \pi \mathrm{/}3$. In addition, we give a $5\mathrm{/}2$-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~$27\mathrm{/}10$.