  We study the uniform $2$-dimensional vector multiple knapsack (2VMK) problem,
a natural variant of multiple knapsack arising in real-world applications such
as virtual machine placement. The input for 2VMK is a set of items, each
associated with a $2$-dimensional weight vector and a positive profit, along
with $m$ $2$-dimensional bins of uniform (unit) capacity in each dimension. The
goal is to find an assignment of a subset of the items to the bins, such that
the total weight of items assigned to a single bin is at most one in each
dimension, and the total profit is maximized.
  Our main result is a $(1- \frac{\ln 2}{2} - \varepsilon)$-approximation
algorithm for 2VMK, for every fixed $\varepsilon > 0$, thus improving the best
known ratio of $(1 - \frac{1}{e}-\varepsilon)$ which follows as a special case
from a result of [Fleischer at al., MOR 2011]. Our algorithm relies on an
adaptation of the Round$\&$Approx framework of [Bansal et al., SICOMP 2010],
originally designed for set covering problems, to maximization problems. The
algorithm uses randomized rounding of a configuration-LP solution to assign
items to $\approx m\cdot \ln 2 \approx 0.693\cdot m$ of the bins, followed by a
reduction to the ($1$-dimensional) Multiple Knapsack problem for assigning
items to the remaining bins.
