Approximations for Allocating Indivisible Items with Concave-Additive Valuations

TL;DR

This paper develops tight approximation algorithms for allocating indivisible items among agents with concave additive valuations, extending prior work and connecting curvature parameters to integrality gaps.

Contribution

It introduces novel approximation methods based on local curvature parameters, providing tight bounds and extending results to Nash welfare maximization.

Findings

Achieves tight multiplicative and additive approximations for the problem.

Establishes a connection between curvature parameters and integrality gaps.

Provides a tatonnement-style interpretation for the allocation process.

Abstract

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.