Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities

TL;DR

This paper extends the approximation algorithms for maximizing Nash Social Welfare from additive valuations to more general separable, piecewise-linear concave utilities, using market equilibria and stable polynomial theories.

Contribution

It introduces two constant factor approximation algorithms for allocating indivisible items under complex utility functions, broadening the scope of prior additive valuation methods.

Findings

Algorithms achieve constant factor approximations

Utilizes market equilibrium and stable polynomial techniques

Advances the design of mechanisms for complex utility functions

Abstract

Recently Cole and Gkatzelis gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash Social Welfare. We give constant factor algorithms for a substantial generalization of their problem -- to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of stable polynomials. In AGT, there is a paucity of methods for the design of mechanisms for the allocation of indivisible goods and the result of Cole and Gkatzelis seemed to be taking a major step towards filling this gap. Our result can be seen as another step in this direction.