Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

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.