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

Nash Welfare Guarantees for Fair and Efficient Coverage