An Asymptotic Lower Bound for Online Vector Bin Packing

TL;DR

This paper establishes a fundamental lower bound on the competitive ratio of any online algorithm for vector bin packing, showing it cannot be asymptotically better than a certain ratio regardless of the instance.

Contribution

It proves an asymptotic lower bound on the competitive ratio for online vector bin packing, extending previous bounds to instances with larger optimal solutions.

Findings

Any online algorithm has an expected number of bins at least proportional to d/ log^3 d times OPT.

The lower bound applies to all randomized algorithms and arbitrary functions q(d).

It rules out the possibility of significantly improved asymptotic competitive ratios for large OPT.

Abstract

We consider the online vector bin packing problem where $n$ items specified by $d$-dimensional vectors must be packed in the fewest number of identical $d$-dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm $A$, there exist instances I, such that $A(I)$, the number of bins used by $A$ to pack $I$, is $\Omega(d/\log^2 d)$ times $OPT(I)$, the minimal number of bins to pack $I$. However in those instances, $OPT(I)$ was only $O(\log d)$, which left open the possibility of improved algorithms with better asymptotic competitive ratio when $OPT(I) \gg d$. We rule this out by showing that for any arbitrary function $q(\cdot)$ and any randomized online algorithm $A$, there exist instances $I$ such that $ E[A(I)] \geq c\cdot d/\log^3d \cdot OPT(I) + q(d)$, for some universal constant $c$.