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We consider the problem of approximating maximum Nash social welfare (NSW)while allocating a set of indivisible items to $n$ agents. The NSW is a popularobjective that provides a balanced tradeoff between the often conflictingrequirements of fairness and efficiency, defined as the weighted geometric meanof agents' valuations. For the symmetric additive case of the problem, whereagents have the same weight with additive valuations, the first constant-factorapproximation algorithm was obtained in 2015. This led to a flurry of workobtaining constant-factor approximation algorithms for the symmetric case undermild generalizations of additive, and $O(n)$-approximation algorithms for moregeneral valuations and for the asymmetric case.  In this paper, we make significant progress towards both symmetric andasymmetric NSW problems. We present the first constant-factor approximationalgorithm for the symmetric case under Rado valuations. Rado valuations form ageneral class of valuation functions that arise from maximum cost independentmatching problems, including as special cases assignment (OXS) valuations andweighted matroid rank functions. Furthermore, our approach also gives the firstconstant-factor approximation algorithm for the asymmetric case under Radovaluations, provided that the maximum ratio between the weights is bounded by aconstant.

Approximating Nash Social Welfare under Rado Valuations