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Computer Science, Computer Science and Game Theory, cs.GT
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 $\sfrac{p}{q}$, for some fixed co-prime numbers $p,q\in
\NN$ 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 $q\geq3$.
