  Packing problems constitute an important class of optimization problems, both
because of their high practical relevance and theoretical appeal. However,
despite the large number of variants that have been studied in the literature,
most packing problems encompass a single tier of capacity restrictions only.
For example, in the Multiple Knapsack Problem, we assign items to multiple
knapsacks such that their capacities are not exceeded. But what if these
knapsacks are partitioned into clusters, each imposing an additional capacity
restriction on the knapsacks contained in that cluster? In this paper, we study
the Maximum Coverage Problem with Cluster Constraints (MCPC), which generalizes
the Maximum Coverage Problem with Knapsack Constraints (MCPK) by incorporating
cluster constraints. Our main contribution is a general LP-based technique to
derive approximation algorithms for cluster capacitated problems. Our technique
allows us to reduce the cluster capacitated problem to the respective original
packing problem. By using an LP-based approximation algorithm for the original
problem, we can then obtain an effective rounding scheme for the problem, which
only loses a small fraction in the approximation guarantee. We apply our
technique to derive approximation algorithms for MCPC. To this aim, we develop
an LP-based $\frac12(1-\frac1e)$-approximation algorithm for MCPK by adapting
the pipage rounding technique. Combined with our reduction technique, we obtain
a $\frac13(1-\frac1e)$-approximation algorithm for MCPC. We also derive
improved results for a special case of MCPC, the Multiple Knapsack Problem with
Cluster Constraints (MKPC). Based on a simple greedy algorithm, our approach
yields a $\frac13$-approximation algorithm. By combining our technique with a
more sophisticated iterative rounding approach, we obtain a
$\frac12$-approximation algorithm for certain special cases of MKPC.
