  We study the $d$-dimensional Vector Bin Packing ($d$VBP) problem, a
generalization of Bin Packing with central applications in resource allocation
and scheduling. In $d$VBP, we are given a set of items, each of which is
characterized by a $d$-dimensional volume vector; the objective is to partition
the items into a minimum number of subsets (bins), such that the total volume
of items in each subset is at most $1$ in each dimension.
  Our main result is an asymptotic approximation algorithm for $d$VBP that
yields a ratio of $(1+\ln d-\chi(d) +\varepsilon)$ for all $d \in \mathbb{N}$
and any $\varepsilon > 0$; here, $\chi(d)$ is some strictly positive function.
This improves upon the best known asymptotic ratio of $ \left(1+ \ln d
+\varepsilon\right)$ due to Bansal, Caprara and Sviridenko (SICOMP 2010) for
any $d >3$. By slightly modifying our algorithm to include an initial matching
phase and applying a tighter analysis we obtain an asymptotic approximation
ratio of $\left(\frac{4}{3}+\varepsilon\right)$ for the special case of $d=2$,
thus substantially improving the previous best ratio of
$\left(\frac{3}{2}+\varepsilon\right)$ due to Bansal, Elias and Khan (SODA
2016).
  Our algorithm iteratively solves a configuration LP relaxation for the
residual instance (from previous iterations) and samples a small number of
configurations based on the solution for the configuration LP. While iterative
rounding was already used by Karmarkar and Karp (FOCS 1982) to establish their
celebrated result for classic (one-dimensional) Bin Packing, iterative
randomized rounding is used here for the first time in the context of (Vector)
Bin Packing. Our results show that iterative randomized rounding is a powerful
tool for approximating $d$VBP, leading to simple algorithms with improved
approximation guarantees.
