Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

Hannaneh Akrami, Bhaskar Ray Chaudhury, Martin Hoefer, Kurt Mehlhorn, Marco Schmalhofer, Golnoosh Shahkarami, Giovanna Varricchio, Quentin Vermande, Ernest van Wijland

Abstract

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

Abstract

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either or , for some fixed co-prime numbers such that . Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of . We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, . We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, . Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever .
Paper Structure (32 sections, 35 theorems, 10 equations, 2 algorithms)

This paper contains 32 sections, 35 theorems, 10 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Our Techniques
  4. Characterizing the allocation of heavy goods:
  5. Allowing heavy goods to be allocated as light goods:
  6. Hardness result:
  7. History of the Paper and Further Related Work
  8. Organization
  9. Preliminaries
  10. Utility Graphs and Utility Profiles
  11. Alternating Paths
  12. Integral Instances
  13. Properties of an Optimal Allocation
  14. Algorithm
  15. Phase 1: Heavy-Only Allocations
  16. ...and 17 more sections

Key Result

Theorem 1

There exists a polynomial-time algorithm that computes a maximum NSW allocation for integral instances, i.e., when $q = 1$ and $p$ is an integer greater than one.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Claim 1
  • Lemma 1
  • Lemma 2: BarmanKV18AAMASHalpernPPS20
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 29 more