We study the complexity of fair division of indivisible goods and consider settings where agents can have nonzero utility for the empty bundle. This is a deviation from a common normalization assumption in the literature, and we show that this inconspicuous change can lead to an increase in complexity: In particular, while an allocation maximizing social welfare by the Nash product is known to be easy to detect in the normalized setting whenever there are as many agents as there are resources, without normalization it can no longer be found in polynomial time, unless P = NP. The same statement also holds for egalitarian social welfare. Moreover, we show that it is NP-complete to decide whether there is an allocation whose Nash product social welfare is above a certain threshold if the number of resources is a multiple of the number of agents. Finally, we consider elitist social welfare and prove that the increase in expressive power by allowing negative coefficients again yields NP-completeness.

中文翻译：

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在不可分割的商品公平分配中放弃正常化的代价

我们研究了不可分割商品公平分割的复杂性，并考虑了代理可以对空束具有非零效用的设置。这与文献中常见的归一化假设背道而驰，我们表明这种不显眼的变化会导致复杂性的增加：特别是，虽然已知通过纳什乘积最大化社会福利的分配很容易在只要有与资源一样多的代理，就可以归一化设置，如果没有归一化，就无法在多项式时间内找到它，除非 P = NP。同样的说法也适用于平等主义的社会福利。此外，我们表明，如果资源数量是代理数量的倍数，则决定是否存在纳什积社会福利高于某个阈值的分配是 NP 完全的。