Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915808260653056 |
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| author | Mehlhorn, Kurt |
| author_facet | Mehlhorn, Kurt |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case Mehlhorn, Kurt Computer Science and Game Theory 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 |
| title | Maximizing Nash Social Welfare in 2-Value Instances: A Simpler Proof for the Half-Integer Case |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2411.06924 |