Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work has shown that when the resource is represented by an interval, a consensus halving with at most $n$ cuts always exists, but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting where the resource consists of a set of items without a linear ordering. When agents have additive utilities, we present a polynomial-time algorithm that computes a consensus halving with at most $n$ cuts, and show that $n$ cuts are almost surely necessary when the agents' utilities are drawn from probabilistic distributions. On the other hand, we show that for a simple class of monotonic utilities, the problem already becomes PPAD-hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus $k$-splitting, where we wish to divide the resource into $k$ parts in possibly unequal ratios, and provide some consequences of our results on the problem of computing small agreeable sets.

中文翻译：

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项目集的共识减半

共识减半是指将资源分成两部分的问题，以便每个代理对这两个部分的价值相等。先前的工作表明，当资源由一个区间表示时，始终存在最多减少 $n$ 的共识减半，但即使对于具有简单估值函数的代理也难以计算。在本文中，我们研究了自然环境中的共识减半，其中资源由一组没有线性排序的项目组成。当代理具有可加效用时，我们提出了一个多项式时间算法，该算法计算最多 n$ 削减的共识减半，并表明当代理的效用来自概率分布时，几乎肯定需要 n$ 削减。另一方面，我们表明，对于一类简单的单调实用程序，问题已经变成了 PPAD-hard。此外，