arXiv 2211.064582022-11-04cs.GT

一个离散有界的局部无嫉妒树蛋糕切割协议

A Discrete and Bounded Locally Envy-Free Cake Cutting Protocol on Trees

Ganesh Ghalme, Xin Huang, Yuka Machino, Nidhi Rathi

备注：: arXiv admin note: substantial text overlap with arXiv:2205.12559

摘要：

我们研究的经典问题的\emph{fairly}划分的异构和可分割的资源-建模为线段$[0,1]$，通常称为\emph{cake}-之间的$n$代理。这项工作考虑了该问题的一个有趣变体，其中代理嵌入在图中。图形化的约束要求每个代理只针对其邻居的份额来评估其分配的份额。给定一个图，我们的目标是有效地找到一个\emph{locally envy-free}分配，其中每个代理都认为她的蛋糕份额至少与她的任何邻居的份额一样多。这项工作的最重要的贡献是一个有界的协议，发现一个本地的羡慕免费分配$n$代理树图上使用$n^{O(n)}$查询下的标准罗伯逊-韦伯（RW）查询模型。我们提出的协议的查询复杂度，虽然是指数级的，但显着提高了Aziz和Mackenzie [AM 16]针对完全图的当前最知名的超指数查询复杂度界限。特别是，我们还表明，如果底层树图的深度最多为2，可以找到一个本地的羡慕免费分配与$O(n^4 \log n)$ RW查询。这是第一个也是唯一一个已知的局部无羡慕蛋糕切割协议，具有多项式查询复杂度的非平凡图结构。有趣的是，我们的离散协议简单且易于理解，而不是[AM 16]的高度复杂的协议。这种简单性可归因于它们的递归性质和使用单个代理作为指定的\emph{cutter}。我们相信，这些结果将有助于我们提高我们的算法理解的可以说是具有挑战性的问题，无羡慕蛋糕切割，通过揭示其查询复杂性和它的关系到底层图结构的瓶颈。

英文摘要：

We study the classic problem of \emph{fairly} dividing a heterogeneous and divisible resource -- modeled as a line segment $[0,1]$ and typically called as a \emph{cake} -- among $n$ agents. This work considers an interesting variant of the problem where agents are embedded on a graph. The graphical constraint entails that each agent evaluates her allocated share only against her neighbors' share. Given a graph, the goal is to efficiently find a \emph{locally envy-free} allocation where every agent values her share of the cake to be at least as much as that of any of her neighbors' share. The most significant contribution of this work is a bounded protocol that finds a locally envy-free allocation among $n$ agents on a tree graph using $n^{O(n)}$ queries under the standard Robertson-Webb (RW) query model. The query complexity of our proposed protocol, though exponential, significantly improves the currently best known hyper-exponential query complexity bound of Aziz and Mackenzie [AM16] for complete graphs. In particular, we also show that if the underlying tree graph has a depth of at most two, one can find a locally envy-free allocation with $O(n^4 \log n)$ RW queries. This is the first and the only known locally envy-free cake cutting protocol with polynomial query complexity for a non-trivial graph structure. Interestingly, our discrete protocols are simple and easy to understand, as opposed to highly involved protocol of [AM16]. This simplicity can be attributed to their recursive nature and the use of a single agent as a designated \emph{cutter}. We believe that these results will help us improve our algorithmic understanding of the arguably challenging problem of envy-free cake-cutting by uncovering the bottlenecks in its query complexity and its relation to the underlying graph structures.

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摘要：

英文摘要：

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