  We study a general allocation setting where agent valuations are concave
additive. In this model, a collection of items must be uniquely distributed
among a set of agents, where each agent-item pair has a specified utility. The
objective is to maximize the sum of agent valuations, each of which is an
arbitrary non-decreasing concave function of the agent's total additive
utility. This setting was studied by Devanur and Jain (STOC 2012) in the online
setting for divisible items. In this paper, we obtain both tight multiplicative
and additive approximations in the offline setting for indivisible items. Our
approximations depend on novel parameters that measure the local
multiplicative/additive curvatures of each agent valuation, which we show
correspond directly to the integrality gap of the natural assignment convex
program of the problem. Furthermore, we extend our additive guarantees to
obtain constant multiplicative approximations for Asymmetric Nash Welfare
Maximization when agents have smooth valuations (Fain et al. EC 2018, Fluschnik
et al. AAAI 2019). This algorithm also yields an interesting tatonnement-style
interpretation, where agents adjust uniform prices and items are assigned
according to maximum weighted bang-per-buck ratios.
