We present a study of a few graph-based problems motivated by fair allocation of resources in a social network. The central role in the paper is played by the following problem: What is the largest number of items we can allocate to the agents in the given social network so that each agent hides at most one item and overall at most items are hidden, and no one envies its neighbors? We show that the problem admits an XP algorithm and is W[1]-hard parameterized by . Moreover, within the running time, we can identify agents that should hide its items and can construct an ordering in which agents should pick items into its bundles to get a desired allocation. Besides this problem, we also consider the existence and verification versions of this problem. In the existence problem, we are given a social network, valuations, a budget, and the goal is to find an allocation without envy. In the verification problem, we are additionally given an allocation, and the goal is to determine if the allocation satisfies the required property.

中文翻译：

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公平分配，社交网络中保留的信息最少

我们提出了一些基于图的问题的研究，这些问题的动机是社交网络中资源的公平分配。本文的核心问题是以下问题：我们可以分配给给定社交网络中的代理的最大项目数是多少，以便每个代理最多隐藏一个项目，并且总体上最多隐藏项目，并且不存在一个人羡慕他的邻居吗？我们证明该问题承认 XP 算法，并且是由 参数化的 W[1]-hard 问题。此外，在运行时间内，我们可以识别应该隐藏其项目的代理，并可以构建一个排序，代理应该将项目挑选到其捆绑包中以获得所需的分配。除了这个问题之外，我们还考虑这个问题的存在性和验证版本。在存在问题中，我们被赋予了一个社交网络、估值、预算，目标是找到一个不带嫉妒的分配。在验证问题中，我们额外给出了一个分配，目标是确定该分配是否满足所需的属性。