Greedy Algorithms for Maximizing Nash Social Welfare

by   Siddharth Barman, et al.

We study the problem of fairly allocating a set of indivisible goods among agents with additive valuations. The extent of fairness of an allocation is measured by its Nash social welfare, which is the geometric mean of the valuations of the agents for their bundles. While the problem of maximizing Nash social welfare is known to be APX-hard in general, we study the effectiveness of simple, greedy algorithms in solving this problem in two interesting special cases. First, we show that a simple, greedy algorithm provides a 1.061-approximation guarantee when agents have identical valuations, even though the problem of maximizing Nash social welfare remains NP-hard for this setting. Second, we show that when agents have binary valuations over the goods, an exact solution (i.e., a Nash optimal allocation) can be found in polynomial time via a greedy algorithm. Our results in the binary setting extend to provide novel, exact algorithms for optimizing Nash social welfare under concave valuations. Notably, for the above mentioned scenarios, our techniques provide a simple alternative to several of the existing, more sophisticated techniques for this problem such as constructing equilibria of Fisher markets or using real stable polynomials.


page 1

page 2

page 3

page 4


Nash Social Welfare for 2-value Instances

We study the problem of allocating a set of indivisible goods among agen...

Fair integer programming under dichotomous preferences

One cannot make truly fair decisions using integer linear programs unles...

Maximizing Social Welfare in Score-Based Social Distance Games

Social distance games have been extensively studied as a coalition forma...

Maximizing Nash Social Welfare in 2-Value Instances: The Half-Integer Case

We consider the problem of maximizing the Nash social welfare when alloc...

Computing Stable Coalitions: Approximation Algorithms for Reward Sharing

Consider a setting where selfish agents are to be assigned to coalitions...

Optimal Nash Equilibria for Bandwidth Allocation

In bandwidth allocation, competing agents wish to transmit data along pa...

Nash Social Welfare in Selfish and Online Load Balancing

In load balancing problems there is a set of clients, each wishing to se...

Please sign up or login with your details

Forgot password? Click here to reset