Maximum Coverage Problem with Group Budget Constraints and Applications

We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets ${\cal S} = \{S_1, S_2, \ldots, S_m\}$ where each set S _i is a subset of a given ground set X. In the maximum coverage problem the goal is to pick k sets from ${\cal S}$ to maximize the cardinality of their union. In the MCG problem ${\cal S}$ is partitioned into groupsG₁, G₂, ..., G_ℓ. The goal is to pick k sets from ${\cal S}$ to maximize the cardinality of their union but with the additional restriction that at most one set be picked from each group. We motivate the study of MCG by pointing out a variety of applications. We show that the greedy algorithm gives a 2-approximation algorithm for this problem which is tight in the oracle model. We also obtain a constant factor approximation algorithm for the cost version of the problem. We then use MCG to obtain the first constant factor approximation algorithms for the following problems: (i) multiple depot k-traveling repairmen problem with covering constraints and (ii) orienteering problem with time windows when the number of time windows is a constant.