Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability
arxiv(2022)
摘要
We study the problem of allocating a set of indivisible goods among a set of
agents with 2-value additive valuations. In this setting, each good is
valued either 1 or pq, for some fixed co-prime numbers p,q∈ such that 1≤ q < p. Our goal is to find an allocation maximizing the
Nash social welfare (), i.e., the geometric mean of the valuations
of the agents. In this work, we give a complete characterization of
polynomial-time tractability of maximization that solely depends on the
values of q.
We start by providing a rather simple polynomial-time algorithm to find a
maximum allocation when the valuation functions are integral, that
is, q=1. We then exploit more involved techniques to get an algorithm
producing a maximum allocation for the half-integral case, that
is, q=2. Finally, we show it is -hard to compute an allocation with
maximum whenever q≥3.
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