Primal-Dual Algorithms for Indivisible Concave Allocation with Bounded Local Curvature.

We study a general allocation setting where agent valuations are specified by an arbitrary set of monotone concave functions. In this model, a collection of items must be uniquely assigned to a set of agents, where all agent-item pairs have a specified bid value. The objective is to maximize the sum of agent valuations, each of which is a nondecreasing concave function of the agent's total spend. This setting was studied by Devanur and Jain (STOC 2012) in the online setting for fractional assignments. In this paper, we generalize the state of the art in the offline setting for indivisible allocations. In particular, for a collection of monotone concave functions and maximum bid value $b$, we define the notion of a local curvature bound $c_\ell\in (0,1]$, which intuitively measures the largest multiplicative gap between any valuation function and a local linear lower bound with $x$-width $b$. For our main contribution, we define an efficient primal-dual algorithm that achieves an approximation of $(1-\epsilon)c_\ell$, and provide a matching $c_\ell$ integrality gap, showing our algorithm is optimal among those that utilize the natural assignment CP. Additionally, we show our techniques have an interesting extension to the Smooth Nash Social Welfare problem (Fain et al. EC 2018, Fluschnik et al. AAAI 2019), which is a variant of NSW that reduces the extent to which the objective penalizes under-allocations by adding a smoothing constant to agent utilities. This objective can be viewed as a sum of logs optimization, which therefore falls in the above setting. However, we also show how our techniques can be extended to obtain an approximation for the canonical product objective by obtaining an additive guarantee for the log version of the problem.