# Approximating Nash Social Welfare under Binary XOS and Binary Subadditive Valuations

We study the problem of allocating indivisible goods among agents in a fair and economically efficient manner. In this context, the Nash social welfare–defined as the geometric mean of agents' valuations for their assigned bundles–stands as a fundamental measure that quantifies the extent of fairness of an allocation. Focusing on instances in which the agents' valuations have binary marginals, we develop essentially tight results for (approximately) maximizing Nash social welfare under two of the most general classes of complement-free valuations, i.e., under binary XOS and binary subadditive valuations. For binary XOS valuations, we develop a polynomial-time algorithm that finds a constant-factor (specifically 288) approximation for the optimal Nash social welfare, in the standard value-oracle model. The allocations computed by our algorithm also achieve constant-factor approximation for social welfare and the groupwise maximin share guarantee. These results imply that–in the case of binary XOS valuations–there necessarily exists an allocation that simultaneously satisfies multiple (approximate) fairness and efficiency criteria. We complement the algorithmic result by proving that Nash social welfare maximization is APX-hard under binary XOS valuations. Furthermore, this work establishes an interesting separation between the binary XOS and binary subadditive settings. In particular, we prove that an exponential number of value queries are necessarily required to obtain even a sub-linear approximation for Nash social welfare under binary subadditive valuations.

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