Approximations for Indivisible Concave Allocations with Applications to Nash Welfare Maximization

We study a general allocation setting where agent valuations are concave additive. In this model, collection of items must be uniquely distributed among set agents, each agent-item pair has specified utility. The objective is to maximize the sum valuations, which an arbitrary non-decreasing function agent's total additive This was studied by Devanur and Jain (STOC 2012) in online for divisible items. paper, we obtain both multiplicative approximations offline indivisible Our depend on novel parameters that measure local multiplicative/additive curvatures valuation, show correspond directly integrality gap natural assignment convex program problem. Furthermore, extend our guarantees constant Asymmetric Nash Welfare Maximization when agents have smooth valuations. algorithm also yields interesting tatonnement-style interpretation, adjust uniform prices assigned according maximum weighted bang-per-buck ratios.