@online{Akrami2207.10949,
TITLE = {Maximizing Nash Social Welfare in 2-Value Instances: The Half-Integer Case},
AUTHOR = {Akrami, Hannaneh and Ray Chaudhury, Bhaskar and Hoefer, Martin and Mehlhorn, Kurt and Schmalhofer, Marco and Shahkarami, Golnoosh and Varricchio, Giovanna and Vermande, Quentin and van Wijland, Ernest},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2207.10949},
EPRINT = {2207.10949},
EPRINTTYPE = {arXiv},
YEAR = {2022},
ABSTRACT = {We consider the problem of maximizing the Nash social welfare when allocating<br>a set $G$ of indivisible goods to a set $N$ of agents. We study instances, in<br>which all agents have 2-value additive valuations: The value of a good $g \in<br>G$ for an agent $i \in N$ is either $1$ or $s$, where $s$ is an odd multiple of<br>$\frac{1}{2}$ larger than one. We show that the problem is solvable in<br>polynomial time. Akrami et at. showed that this problem is solvable in<br>polynomial time if $s$ is integral and is NP-hard whenever $s = \frac{p}{q}$,<br>$p \in \mathbb{N}$ and $q\in \mathbb{N}$ are co-prime and $p > q \ge 3$. For<br>the latter situation, an approximation algorithm was also given. It obtains an<br>approximation ratio of at most $1.0345$. Moreover, the problem is APX-hard,<br>with a lower bound of $1.000015$ achieved at $\frac{p}{q} = \frac{5}{4}$. The<br>case $q = 2$ and odd $p$ was left open.<br> In the case of integral $s$, the problem is separable in the sense that the<br>optimal allocation of the heavy goods (= value $s$ for some agent) is<br>independent of the number of light goods (= value $1$ for all agents). This<br>leads to an algorithm that first computes an optimal allocation of the heavy<br>goods and then adds the light goods greedily. This separation no longer holds<br>for $s = \frac{3}{2}$; a simple example is given in the introduction. Thus an<br>algorithm has to consider heavy and light goods together. This complicates<br>matters considerably. Our algorithm is based on a collection of improvement<br>rules that transfers any allocation into an optimal allocation and exploits a<br>connection to matchings with parity constraints.<br>},
}