An unceasing problem of our prevailing society is the fair division of goods. The problem of proportional cake cutting focuses on dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share. In this article, we investigate the problem of proportional division with unequal shares, where each player is entitled to receive a predetermined portion of the cake. Our main contribution is threefold. First we present a protocol for integer demands, which delivers a proportional solution in fewer queries than all known protocols. By giving a matching lower bound, we then show that our protocol is asymptotically the fastest possible. Finally, we turn to irrational demands and solve the proportional cake cutting problem by reducing it to the same problem with integer demands only. All results remain valid in a highly general cake cutting model, which can be of independent interest.

中文翻译：

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不等份蛋糕切割的复杂性

我们当前社会的一个持续存在的问题是商品的公平分配。比例蛋糕切割问题的重点是如何将一种异质且可分割的资源蛋糕，n玩家根据自己的度量函数来评估棋子。目标是为每个玩家分配一个蛋糕中不一定相连的部分，玩家对蛋糕的评价至少与她的比例份额一样多。在本文中，我们研究了不均等份额的比例分配问题，其中每个玩家都有权获得预定部分的蛋糕。我们的主要贡献是三方面的。首先，我们提出了一个整数需求协议，它在比所有已知协议更少的查询中提供了比例解决方案。通过给出匹配的下限，我们可以证明我们的协议是渐近最快的。最后，我们转向非理性需求，并通过将其简化为仅具有整数需求的相同问题来解决比例蛋糕切割问题。