  In [Math. Oper. Res., 2011], Fleischer et al. introduced a powerful technique
for solving the generic class of separable assignment problems (SAP), in which
a set of items of given values and weights needs to be packed into a set of
bins subject to separable assignment constraints, so as to maximize the total
value. The approach of Fleischer at al. relies on solving a configuration LP
and sampling a configuration for each bin independently based on the LP
solution. While there is a SAP variant for which this approach yields the best
possible approximation ratio, for various special cases, there are
discrepancies between the approximation ratios obtained using the above
approach and the state-of-the-art approximations. This raises the following
natural question: Can we do better by iteratively solving the configuration LP
and sampling a few bins at a time?
  To assess the potential gain from iterative randomized rounding, we consider
as a case study one interesting SAP variant, namely, Uniform Cardinality
Constrained Multiple Knapsack, for which we answer this question affirmatively.
The input is a set of items, each has a value and a weight, and a set of
uniform capacity bins. The goal is to assign a subset of the items of maximum
total value to the bins such that $(i)$ the capacity of any bin is not
exceeded, and $(ii)$ the number of items assigned to each bin satisfies a given
cardinality constraint. While the technique of Fleischer et al. yields a
$\left(1-\frac{1}{e}\right)$-approximation for the problem, we show that
iterative randomized rounding leads to an efficient polynomial time
approximation scheme (EPTAS), thus essentially resolving the complexity status
of the problem. Our analysis of iterative randomized rounding can be useful for
solving other SAP variants.
