  We study coverage problems in which, for a set of agents and a given
threshold $T$, the goal is to select $T$ subsets (of the agents) that, while
satisfying combinatorial constraints, achieve fair and efficient coverage among
the agents. In this setting, the valuation of each agent is equated to the
number of selected subsets that contain it, plus one. The current work utilizes
the Nash social welfare function to quantify the extent of fairness and
collective efficiency. We develop a polynomial-time $\left(18 + o(1)
\right)$-approximation algorithm for maximizing Nash social welfare in coverage
instances. Our algorithm applies to all instances wherein, for the underlying
combinatorial constraints, there exists an FPTAS for weight maximization. We
complement the algorithmic result by proving that Nash social welfare
maximization is APX-hard in coverage instances.
