Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case

Kurt Mehlhorn

Abstract

A set of $m$ indivisible goods is to be allocated to a set of $n$ agents. Each agent $i$ has an additive valuation function $v_i$ over goods. The value of a good $g$ for agent $i$ is either $1$ or $s$, where $s$ is a fixed rational number greater than one, and the value of a bundle of goods is the sum of the values of the goods in the bundle. An \emph{allocation} $X$ is a partition of the goods into bundles $X_1$, \ldots, $X_n$, one for each agent. The \emph{Nash Social Welfare} ($\NSW$) of an allocation $X$ is defined as \[ \NSW(X) = \left( \prod_i v_i(X_i) \right)^{\sfrac{1}{n}}.\] The \emph{$\NSW$-allocation} maximizes the Nash Social Welfare. In~\cite{NSW-twovalues-halfinteger} it was shown that the $\NSW$-allocation can be computed in polynomial time, if $s$ is an integer or a half-integer, and that the problem is NP-complete otherwise. The proof for the half-integer case is quite involved. In this note we give a simpler and shorter proof

Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case

Abstract

A set of indivisible goods is to be allocated to a set of agents. Each agent has an additive valuation function over goods. The value of a good for agent is either or , where is a fixed rational number greater than one, and the value of a bundle of goods is the sum of the values of the goods in the bundle. An \emph{allocation} is a partition of the goods into bundles , \ldots, , one for each agent. The \emph{Nash Social Welfare} () of an allocation is defined as The \emph{-allocation} maximizes the Nash Social Welfare. In~\cite{NSW-twovalues-halfinteger} it was shown that the -allocation can be computed in polynomial time, if is an integer or a half-integer, and that the problem is NP-complete otherwise. The proof for the half-integer case is quite involved. In this note we give a simpler and shorter proof

Paper Structure

This paper contains 8 sections, 17 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Related Work:
  3. The Algorithm
  4. The LexMin Allocation of the Heavy Goods
  5. Assignment of the Light Goods and Optimization of the Small Bundles
  6. Case $j \in A_0 \cup A_1$:
  7. Case $j \in (O_0 \cup O_1) \cap A_{1/2}$:
  8. The Algorithmics of Optimizing the Small Bundles

Key Result

Lemma 1

For any $\ell$, agents in $R_\ell \cup R'_\ell$ own either $\ell$ or $\ell - 1$ heavy goods. Each of them could own $\ell$ goods via a transfer from another agent in the set. For any $k$, all heavy goods assigned to the agents in $\cup_{\ell \ge k} R_\ell \cup R'_\ell$ must be assigned to them.

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6: Lemma 26 from NSW-twovalues-halfinteger
  • Lemma 7: Global Accounting of Light Goods
  • Lemma 8
  • Lemma 9: Lemmas 14 and 15 in NSW-twovalues-halfinteger
  • Lemma 10
  • ...and 7 more