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

We consider the problem of maximizing the Nash social welfare when allocating a set $G$ of indivisible goods to a set $N$ of agents. We study instances, in which all agents have 2-value additive valuations: The value of a good $g \in G$ for an agent $i \in N$ is either $1$ or $s$, where $s$ is an odd multiple of $\frac{1}{2}$ larger than one. We show that the problem is solvable in polynomial time. Akrami et at. showed that this problem is solvable in polynomial time if $s$ is integral and is NP-hard whenever $s = \frac{p}{q}$, $p \in \mathbb{N}$ and $q\in \mathbb{N}$ are co-prime and $p > q \ge 3$. For the latter situation, an approximation algorithm was also given. It obtains an approximation ratio of at most $1.0345$. Moreover, the problem is APX-hard, with a lower bound of $1.000015$ achieved at $\frac{p}{q} = \frac{5}{4}$. The case $q = 2$ and odd $p$ was left open. In the case of integral $s$, the problem is separable in the sense that the optimal allocation of the heavy goods (= value $s$ for some agent) is independent of the number of light goods (= value $1$ for all agents). This leads to an algorithm that first computes an optimal allocation of the heavy goods and then adds the light goods greedily. This separation no longer holds for $s = \frac{3}{2}$; a simple example is given in the introduction. Thus an algorithm has to consider heavy and light goods together. This complicates matters considerably. Our algorithm is based on a collection of improvement rules that transfers any allocation into an optimal allocation and exploits a connection to matchings with parity constraints.